messages listlengths 1 1 | ground_truth listlengths 15 15 | dataset stringclasses 1 value | id stringlengths 5 8 | difficulty int64 3 3 | timestamps listlengths 15 15 | scene_config dict | tolerance float64 50 50 | prompt_setting stringclasses 1 value |
|---|---|---|---|---|---|---|---|---|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (706, 854), initial velocity (736.03, -460.43) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (715, 630), initial velocity (160.0, -949.15) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (955, 685), initial velocity (-393.01, 911.19) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (769, 1016), initial velocity (188.32, -888.7) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (887, 791), initial velocity (147.05, -792.19) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[779.61, 807.96], [730.99, 535.08], [947.27, 757.88], [787.82, 927.13], [838.54, 748.24]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[853.22, 761.91], [753.94, 461.57], [984.14, 805.03], [806.65, 838.27], [700.89, 756.98]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[927.01, 715.75], [822.83, 529.12], [1021.1, 852.29], [825.52, 749.18], [562.9, 765.75]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1000.62, 669.71], [891.56, 596.5], [974.02, 838.43], [844.34, 660.31], [488.0, 774.49]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[990.96, 684.15], [960.29, 663.89], [917.8, 817.92], [863.16, 571.44], [625.65, 783.23]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[986.75, 802.24], [948.06, 650.32], [861.57, 797.42], [881.99, 482.57], [763.29, 791.97]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[975.48, 915.61], [933.81, 634.72], [866.28, 741.72], [829.23, 445.72], [779.08, 871.06]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[862.49, 955.12], [919.55, 619.12], [898.56, 670.12], [749.74, 469.56], [739.75, 981.97]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[766.04, 985.07], [905.29, 555.68], [930.83, 635.33], [682.26, 530.39], [702.95, 1002.38]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[770.22, 956.92], [891.03, 476.8], [963.11, 612.43], [614.78, 591.23], [535.48, 921.75]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[774.39, 928.77], [840.66, 424.15], [982.86, 598.63], [547.3, 652.07], [527.16, 785.49]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[778.57, 900.62], [804.31, 491.67], [951.12, 622.24], [479.83, 712.91], [676.55, 649.24]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[782.73, 872.54], [808.52, 496.2], [919.46, 645.79], [520.73, 773.6], [765.58, 617.22]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[786.91, 844.39], [864.61, 452.5], [887.73, 669.4], [588.21, 834.44], [749.69, 767.25]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[791.08, 816.24], [872.79, 587.15], [855.99, 693.0], [655.69, 895.28], [733.8, 917.27]]) <= 50.0"
] | ballsim | G_3_47 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
208,
80,
129
],
"density": 1,
"position": [
706,
854
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
736.03,
-460.43
]
},
{
"color": [
172,
231,
90
],
"density": 1,
"position": [
715,
630
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
160,
-949.15
]
},
{
"color": [
247,
66,
247
],
"density": 1,
"position": [
955,
685
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-393.01,
911.19
]
},
{
"color": [
63,
193,
224
],
"density": 1,
"position": [
769,
1016
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
188.32,
-888.7
]
},
{
"color": [
65,
170,
96
],
"density": 1,
"position": [
887,
791
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
147.05,
-792.19
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 47,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4366641
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (840, 840), initial velocity (3.38, 811.52) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (483, 796), initial velocity (777.67, -383.74) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (624, 851), initial velocity (878.64, -56.78) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (871, 804), initial velocity (325.82, -897.19) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (691, 619), initial velocity (266.17, 766.34) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[840.34, 921.15], [560.76, 757.62], [711.87, 845.31], [903.59, 714.28], [717.61, 695.64]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[840.68, 1002.3], [638.52, 719.24], [799.73, 839.62], [936.19, 624.56], [744.22, 772.27]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[841.03, 1027.8], [716.48, 680.77], [887.82, 833.92], [968.86, 534.61], [770.9, 849.1]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[841.37, 946.64], [794.23, 642.39], [975.68, 828.23], [892.53, 507.77], [797.51, 925.73]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[841.71, 865.49], [871.99, 604.01], [1062.89, 822.17], [798.53, 491.13], [824.12, 1002.37]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[842.05, 784.34], [949.75, 565.63], [1023.87, 743.24], [704.54, 474.49], [850.73, 1030.51]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[842.39, 703.19], [950.82, 571.54], [984.86, 664.31], [610.54, 457.85], [877.35, 953.88]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[840.62, 623.26], [880.64, 618.59], [945.85, 585.38], [606.1, 492.92], [903.96, 877.24]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[725.73, 608.63], [913.96, 605.87], [906.83, 506.44], [667.51, 565.99], [930.57, 800.61]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[738.83, 687.0], [947.29, 593.15], [867.82, 457.63], [600.93, 546.07], [957.18, 723.97]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[781.04, 786.51], [980.61, 580.43], [828.81, 536.56], [556.51, 534.6], [983.79, 647.33]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[823.25, 886.02], [966.78, 565.07], [789.79, 615.49], [639.91, 596.93], [1010.4, 609.51]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[865.36, 985.28], [891.13, 559.78], [750.88, 694.22], [723.1, 659.1], [1013.55, 618.64]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[907.57, 1026.66], [815.3, 554.48], [711.86, 773.15], [806.5, 721.43], [996.43, 639.49]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[949.78, 927.15], [739.47, 549.19], [672.85, 852.08], [889.9, 783.76], [979.32, 660.34]]) <= 50.0"
] | ballsim | G_3_1018 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
213,
218,
60
],
"density": 1,
"position": [
840,
840
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
3.38,
811.52
]
},
{
"color": [
217,
231,
208
],
"density": 1,
"position": [
483,
796
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
777.67,
-383.74
]
},
{
"color": [
251,
119,
226
],
"density": 1,
"position": [
624,
851
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
878.64,
-56.78
]
},
{
"color": [
223,
140,
200
],
"density": 1,
"position": [
871,
804
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
325.82,
-897.19
]
},
{
"color": [
74,
247,
226
],
"density": 1,
"position": [
691,
619
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
266.17,
766.34
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1018,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1648766
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (741, 618), initial velocity (-954.48, -228.51) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (944, 556), initial velocity (-869.99, -206.29) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (837, 622), initial velocity (776.64, -319.66) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (816, 742), initial velocity (512.93, 669.68) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (695, 941), initial velocity (868.93, -186.54) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (708, 1020), initial velocity (215.64, -937.27) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[645.54, 595.15], [857.01, 535.37], [914.66, 590.04], [867.29, 808.97], [781.89, 922.35], [729.56, 926.27]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[550.08, 572.3], [770.03, 514.74], [992.32, 558.08], [918.59, 875.94], [868.78, 903.7], [751.12, 832.55]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[539.05, 598.27], [682.82, 494.06], [928.45, 607.87], [1005.27, 914.26], [909.35, 911.87], [772.73, 738.59]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[606.56, 669.51], [595.83, 473.43], [861.94, 659.14], [921.63, 832.94], [929.63, 931.67], [794.28, 644.86]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[674.06, 740.75], [595.47, 502.82], [795.44, 710.41], [837.99, 751.61], [949.92, 951.47], [815.84, 551.14]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[741.57, 811.99], [656.83, 567.86], [676.17, 731.22], [802.62, 698.17], [970.21, 971.27], [837.4, 457.41]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[809.07, 883.23], [718.18, 632.9], [536.28, 740.13], [786.11, 655.61], [953.46, 969.69], [858.96, 521.45]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[876.58, 954.47], [779.53, 697.94], [399.6, 750.9], [769.61, 613.06], [926.17, 962.02], [880.51, 615.18]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[838.8, 1025.71], [840.88, 762.98], [475.61, 868.56], [753.11, 570.5], [956.4, 954.36], [902.07, 708.91]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[783.7, 1017.9], [902.02, 828.23], [553.27, 985.26], [736.6, 527.95], [989.08, 942.64], [923.97, 802.29]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[728.6, 946.66], [936.42, 920.23], [630.93, 1009.5], [720.1, 485.4], [975.87, 904.43], [987.05, 854.49]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[728.59, 835.41], [944.99, 997.31], [678.49, 914.67], [703.59, 442.84], [962.67, 866.23], [996.71, 875.84]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[813.63, 662.65], [848.35, 1013.48], [679.46, 853.83], [687.13, 484.75], [907.56, 828.11], [996.92, 847.37]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[898.88, 489.45], [751.48, 1029.69], [680.44, 792.85], [670.63, 527.3], [849.45, 789.91], [1002.37, 818.83]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[984.14, 568.89], [654.6, 1045.9], [681.42, 731.86], [654.12, 569.85], [791.35, 751.7], [1007.81, 790.29]]) <= 50.0"
] | ballsim | G_3_288 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
174,
218,
251
],
"density": 1,
"position": [
741,
618
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-954.48,
-228.51
]
},
{
"color": [
170,
234,
86
],
"density": 1,
"position": [
944,
556
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-869.99,
-206.29
]
},
{
"color": [
230,
206,
133
],
"density": 1,
"position": [
837,
622
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
776.64,
-319.66
]
},
{
"color": [
81,
109,
76
],
"density": 1,
"position": [
816,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
512.93,
669.68
]
},
{
"color": [
232,
212,
120
],
"density": 1,
"position": [
695,
941
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
868.93,
-186.54
]
},
{
"color": [
123,
89,
177
],
"density": 1,
"position": [
708,
1020
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
215.64,
-937.27
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 288,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4399566
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (731, 682), initial velocity (617.62, 602.99) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (727, 738), initial velocity (-258.54, 878.51) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (740, 494), initial velocity (-28.22, 865.51) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (854, 970), initial velocity (-899.16, 252.69) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (787, 817), initial velocity (708.96, -455.48) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[792.77, 742.3], [701.15, 825.84], [737.17, 580.55], [764.08, 995.27], [857.9, 771.44]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[854.54, 802.61], [675.29, 913.68], [734.34, 667.1], [674.17, 1020.54], [928.8, 725.89]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[916.46, 863.06], [683.85, 967.26], [731.5, 753.86], [587.69, 979.46], [999.87, 680.22]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[978.22, 923.36], [730.95, 982.16], [728.66, 840.41], [475.82, 955.63], [1005.87, 681.81]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[902.84, 884.02], [778.04, 997.05], [725.83, 926.96], [559.78, 902.46], [940.63, 735.16]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[826.4, 843.91], [825.14, 1011.94], [723.0, 1013.5], [662.12, 849.28], [875.4, 788.5]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[788.06, 823.94], [872.23, 1026.84], [754.13, 995.54], [731.79, 777.24], [814.52, 846.21]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[794.62, 805.51], [884.9, 1016.72], [768.76, 957.82], [722.41, 659.57], [851.8, 921.31]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[801.19, 787.08], [856.17, 976.54], [708.24, 995.27], [713.04, 541.89], [978.44, 907.05]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[807.76, 768.65], [827.43, 936.35], [655.94, 1007.41], [728.6, 500.97], [969.15, 794.06]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[814.32, 750.21], [798.7, 896.17], [628.98, 941.54], [790.17, 601.67], [943.56, 669.23]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[820.89, 731.78], [769.96, 855.98], [602.03, 875.67], [851.74, 702.38], [917.98, 544.4]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[827.44, 713.39], [741.3, 815.9], [575.15, 809.97], [913.16, 802.84], [856.7, 445.86]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[834.01, 694.96], [712.56, 775.71], [548.19, 744.1], [974.73, 903.55], [786.67, 508.5]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[840.58, 676.53], [683.83, 735.53], [521.24, 678.23], [881.01, 891.43], [733.99, 624.52]]) <= 50.0"
] | ballsim | G_3_617 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
95,
115,
132
],
"density": 1,
"position": [
731,
682
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
617.62,
602.99
]
},
{
"color": [
118,
115,
227
],
"density": 1,
"position": [
727,
738
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-258.54,
878.51
]
},
{
"color": [
67,
159,
172
],
"density": 1,
"position": [
740,
494
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-28.22,
865.51
]
},
{
"color": [
190,
102,
147
],
"density": 1,
"position": [
854,
970
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-899.16,
252.69
]
},
{
"color": [
138,
73,
67
],
"density": 1,
"position": [
787,
817
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
708.96,
-455.48
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 617,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4495167
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (660, 610), initial velocity (-25.18, -890.83) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (692, 733), initial velocity (-164.21, 958.8) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (967, 867), initial velocity (-642.09, 560.99) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (619, 854), initial velocity (876.43, -240.94) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (690, 862), initial velocity (727.08, -384.51) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[657.49, 520.91], [683.92, 803.2], [902.79, 923.1], [691.38, 876.89], [762.71, 823.55]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[654.97, 458.67], [699.44, 800.79], [838.58, 979.21], [720.58, 1032.7], [835.41, 785.1]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[652.45, 547.98], [715.01, 798.36], [774.21, 1035.45], [749.85, 925.96], [908.3, 746.55]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[649.94, 637.07], [730.54, 795.95], [710.0, 1017.94], [779.05, 770.15], [981.0, 708.1]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[647.42, 726.16], [746.06, 793.53], [645.79, 961.84], [808.25, 614.34], [1053.71, 669.64]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[644.91, 815.24], [761.59, 791.11], [581.59, 905.74], [837.45, 458.53], [991.37, 709.16]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[642.39, 904.33], [777.12, 788.7], [517.38, 849.63], [866.65, 582.42], [921.72, 752.91]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[639.88, 993.42], [792.65, 786.28], [453.17, 793.53], [895.85, 738.23], [852.07, 796.66]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[637.36, 1027.01], [789.53, 770.32], [408.93, 748.96], [925.05, 894.05], [799.47, 852.8]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[634.85, 937.92], [769.75, 742.27], [489.61, 776.52], [920.68, 1030.48], [762.12, 920.01]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[632.33, 848.84], [749.98, 714.22], [570.3, 804.08], [771.14, 1031.75], [724.76, 987.22]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[696.36, 826.29], [730.2, 686.17], [584.45, 765.11], [770.91, 1049.53], [612.77, 1011.33]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[793.53, 837.11], [710.48, 658.19], [565.26, 692.92], [802.67, 997.57], [589.13, 966.05]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[890.94, 847.95], [690.7, 630.14], [546.02, 620.56], [834.5, 945.5], [666.01, 862.58]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[988.35, 858.78], [670.93, 602.08], [526.78, 548.2], [866.34, 893.42], [742.89, 759.11]]) <= 50.0"
] | ballsim | G_3_975 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
66,
248,
252
],
"density": 1,
"position": [
660,
610
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-25.18,
-890.83
]
},
{
"color": [
214,
177,
88
],
"density": 1,
"position": [
692,
733
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-164.21,
958.8
]
},
{
"color": [
87,
103,
132
],
"density": 1,
"position": [
967,
867
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-642.09,
560.99
]
},
{
"color": [
240,
224,
151
],
"density": 1,
"position": [
619,
854
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
876.43,
-240.94
]
},
{
"color": [
163,
217,
127
],
"density": 1,
"position": [
690,
862
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
727.08,
-384.51
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 975,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5203587
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (685, 561), initial velocity (786.33, 487.27) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (566, 831), initial velocity (-688.12, -615.5) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (791, 665), initial velocity (-343.41, 785.7) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (733, 760), initial velocity (-705.8, 651.16) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (798, 939), initial velocity (937.71, -252.19) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[763.64, 609.73], [497.18, 769.45], [756.65, 743.56], [662.42, 825.11], [891.77, 913.78]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[842.28, 658.46], [428.37, 707.9], [722.3, 822.13], [591.84, 890.22], [985.55, 888.56]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[921.11, 707.31], [502.72, 728.95], [687.86, 900.89], [521.08, 955.5], [1019.6, 828.66]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[959.37, 715.66], [590.43, 757.75], [653.51, 979.45], [601.62, 933.36], [1028.52, 768.81]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[904.26, 630.65], [678.15, 786.56], [619.16, 1056.84], [693.29, 904.8], [1067.03, 759.24]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[849.16, 545.64], [765.87, 815.37], [584.81, 978.28], [784.97, 876.23], [1007.13, 692.85]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[794.06, 460.63], [841.84, 832.43], [550.46, 899.71], [885.69, 856.72], [947.24, 626.47]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[738.96, 514.88], [899.36, 831.07], [516.11, 821.15], [1000.61, 851.4], [887.35, 560.09]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[683.85, 599.89], [956.88, 829.7], [481.77, 742.58], [1016.53, 788.91], [827.46, 493.7]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[628.75, 684.9], [1014.4, 828.33], [447.43, 664.02], [963.7, 686.71], [767.58, 461.23]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[573.65, 769.91], [1054.87, 817.12], [528.48, 652.07], [910.87, 584.51], [707.69, 527.62]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[518.55, 854.91], [1027.28, 766.61], [613.68, 642.52], [858.03, 482.32], [647.8, 594.01]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[525.02, 904.24], [999.76, 716.22], [662.85, 743.28], [805.33, 504.75], [607.64, 599.98]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[626.19, 899.01], [972.17, 665.71], [701.28, 877.73], [752.5, 606.95], [573.31, 587.69]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[727.36, 893.79], [944.59, 615.2], [739.71, 1012.18], [699.67, 709.15], [538.99, 575.41]]) <= 50.0"
] | ballsim | G_3_583 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
125,
71,
135
],
"density": 1,
"position": [
685,
561
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
786.33,
487.27
]
},
{
"color": [
53,
98,
208
],
"density": 1,
"position": [
566,
831
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-688.12,
-615.5
]
},
{
"color": [
135,
176,
122
],
"density": 1,
"position": [
791,
665
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-343.41,
785.7
]
},
{
"color": [
121,
139,
78
],
"density": 1,
"position": [
733,
760
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-705.8,
651.16
]
},
{
"color": [
152,
118,
72
],
"density": 1,
"position": [
798,
939
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
937.71,
-252.19
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 583,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1146209
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (897, 703), initial velocity (18.47, 829.7) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (790, 987), initial velocity (-858.91, 352.24) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (797, 728), initial velocity (-495.38, -639.92) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (929, 929), initial velocity (-795.96, 449.63) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (607, 811), initial velocity (-463.9, -817.71) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (955, 881), initial velocity (-753.07, -378.07) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[898.86, 785.96], [704.11, 1022.23], [747.46, 664.01], [849.41, 973.97], [560.61, 729.24], [879.68, 843.18]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[900.71, 868.92], [648.7, 963.65], [697.93, 600.02], [769.82, 1018.94], [514.23, 647.47], [804.37, 805.37]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[902.57, 952.08], [599.8, 884.47], [648.27, 535.87], [710.66, 1000.54], [467.72, 565.51], [728.86, 767.45]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[895.87, 1028.83], [551.02, 805.49], [613.6, 517.64], [672.69, 917.39], [530.22, 563.57], [653.54, 729.64]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[816.41, 1052.71], [502.24, 726.51], [641.61, 576.39], [634.73, 834.23], [585.34, 624.6], [578.22, 691.82]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[755.65, 1018.98], [459.81, 647.53], [716.76, 600.9], [596.39, 752.24], [593.53, 719.23], [502.91, 654.0]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[705.38, 952.94], [509.23, 548.65], [791.9, 625.42], [496.27, 860.35], [635.48, 710.0], [474.35, 643.19]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[655.11, 886.9], [631.4, 629.12], [867.05, 649.93], [517.1, 968.45], [677.42, 700.77], [484.05, 634.35]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[604.84, 820.86], [714.11, 641.25], [942.2, 674.44], [659.11, 947.6], [740.92, 728.87], [509.2, 625.52]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[554.57, 754.82], [760.11, 589.79], [1017.34, 698.95], [803.65, 918.98], [824.49, 791.71], [534.35, 616.68]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[504.3, 688.78], [806.1, 538.32], [1083.61, 729.91], [948.18, 890.37], [908.06, 854.55], [559.49, 607.84]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[479.22, 622.74], [852.1, 486.86], [1055.51, 788.45], [1002.49, 796.21], [980.81, 909.54], [584.64, 599.0]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[529.36, 556.87], [869.87, 455.94], [980.5, 812.73], [985.08, 650.26], [895.43, 849.63], [609.72, 590.19]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[588.76, 518.95], [806.71, 483.77], [905.3, 837.07], [913.42, 543.32], [809.84, 789.57], [634.87, 581.35]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[668.23, 534.18], [743.56, 511.6], [830.11, 861.42], [779.65, 481.51], [724.24, 729.51], [660.02, 579.16]]) <= 50.0"
] | ballsim | G_3_803 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
156,
109,
105
],
"density": 1,
"position": [
897,
703
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
18.47,
829.7
]
},
{
"color": [
69,
113,
97
],
"density": 1,
"position": [
790,
987
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-858.91,
352.24
]
},
{
"color": [
123,
226,
151
],
"density": 1,
"position": [
797,
728
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-495.38,
-639.92
]
},
{
"color": [
59,
143,
90
],
"density": 1,
"position": [
929,
929
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-795.96,
449.63
]
},
{
"color": [
160,
90,
184
],
"density": 1,
"position": [
607,
811
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-463.9,
-817.71
]
},
{
"color": [
61,
234,
254
],
"density": 1,
"position": [
955,
881
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-753.07,
-378.07
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 803,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3732039
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (799, 482), initial velocity (-495.78, 861.07) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (728, 747), initial velocity (-428.87, -729.35) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (862, 598), initial velocity (-254.63, 964.54) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (915, 640), initial velocity (839.46, -114.4) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (573, 642), initial velocity (936.94, 15.87) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[749.42, 568.11], [685.1, 674.08], [836.54, 694.46], [998.94, 628.55], [666.7, 643.59]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[699.83, 654.22], [705.84, 711.32], [811.07, 790.92], [1000.95, 664.4], [696.77, 534.98]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[650.12, 740.54], [729.44, 753.51], [785.54, 887.62], [948.94, 731.53], [724.11, 463.89]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[600.54, 826.65], [752.97, 795.6], [760.08, 984.08], [897.06, 798.5], [751.38, 577.33]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[550.95, 912.76], [776.51, 837.69], [734.62, 1034.33], [845.18, 865.47], [778.65, 690.79]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[557.36, 966.54], [800.04, 879.78], [709.15, 937.87], [793.3, 932.44], [805.92, 804.24]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[656.72, 966.66], [812.79, 940.56], [683.69, 841.41], [741.42, 999.4], [843.97, 899.0]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[756.09, 966.79], [806.37, 1034.55], [658.23, 744.95], [689.54, 1048.47], [901.2, 960.55]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[855.45, 966.91], [799.95, 986.31], [632.76, 648.49], [637.66, 981.49], [942.62, 1012.97]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[954.82, 967.03], [793.53, 892.32], [607.3, 552.03], [585.78, 914.53], [860.69, 994.17]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[942.33, 902.57], [787.11, 798.32], [581.83, 455.57], [533.9, 847.56], [778.75, 975.38]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[892.55, 816.59], [780.69, 704.33], [556.37, 526.03], [482.02, 780.59], [696.82, 956.58]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[842.89, 730.81], [774.28, 610.57], [530.97, 622.25], [430.27, 713.79], [615.09, 937.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[793.11, 644.83], [767.86, 516.57], [528.45, 705.46], [469.74, 726.06], [533.16, 919.03]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[743.33, 558.84], [761.44, 462.55], [621.86, 733.29], [434.81, 806.13], [497.01, 873.8]]) <= 50.0"
] | ballsim | G_3_537 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
185,
192,
117
],
"density": 1,
"position": [
799,
482
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-495.78,
861.07
]
},
{
"color": [
109,
161,
134
],
"density": 1,
"position": [
728,
747
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-428.87,
-729.35
]
},
{
"color": [
75,
172,
114
],
"density": 1,
"position": [
862,
598
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-254.63,
964.54
]
},
{
"color": [
116,
182,
112
],
"density": 1,
"position": [
915,
640
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
839.46,
-114.4
]
},
{
"color": [
245,
158,
124
],
"density": 1,
"position": [
573,
642
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
936.94,
15.87
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 537,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 9273495
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (662, 711), initial velocity (-848.6, 370.0) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (496, 874), initial velocity (869.61, -240.7) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (672, 511), initial velocity (825.38, -322.28) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (715, 748), initial velocity (-589.51, -767.61) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (670, 664), initial velocity (-94.33, 869.36) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[577.14, 748.01], [582.96, 849.93], [754.54, 478.77], [726.35, 741.54], [601.44, 691.8]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[492.27, 785.02], [669.93, 825.86], [837.09, 446.55], [783.21, 780.58], [494.6, 681.33]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[451.38, 796.62], [757.11, 801.72], [919.84, 476.28], [840.21, 819.71], [471.79, 719.49]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[473.81, 831.79], [844.07, 777.65], [947.15, 540.39], [897.07, 858.75], [562.72, 757.53]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[508.71, 896.14], [931.03, 753.58], [933.8, 627.99], [953.94, 897.79], [670.87, 765.76]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[590.96, 933.16], [1017.99, 729.51], [920.45, 715.59], [990.29, 924.99], [779.03, 773.98]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[673.21, 970.17], [1066.41, 727.69], [910.55, 805.7], [928.06, 895.28], [884.27, 780.1]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[755.46, 1007.18], [1002.1, 790.97], [954.55, 914.28], [853.84, 877.54], [954.24, 760.59]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[837.71, 1044.19], [937.77, 854.25], [927.76, 956.71], [763.59, 875.83], [1024.21, 741.08]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[919.96, 1033.64], [873.44, 917.54], [817.58, 951.0], [673.33, 874.12], [1085.78, 726.43]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[942.78, 962.32], [809.11, 980.82], [707.4, 945.28], [583.07, 872.41], [1033.87, 777.26]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[933.7, 872.57], [744.77, 1044.1], [597.22, 939.57], [492.81, 870.7], [981.99, 828.09]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[885.05, 783.05], [680.6, 1007.64], [508.1, 921.87], [494.18, 816.24], [951.86, 878.79]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[820.6, 693.3], [616.27, 944.36], [558.24, 823.6], [537.83, 737.24], [930.23, 929.62]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[756.14, 603.56], [551.94, 881.07], [608.39, 725.34], [581.48, 658.23], [908.6, 980.45]]) <= 50.0"
] | ballsim | G_3_700 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
94,
198,
93
],
"density": 1,
"position": [
662,
711
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-848.6,
370
]
},
{
"color": [
195,
96,
236
],
"density": 1,
"position": [
496,
874
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
869.61,
-240.7
]
},
{
"color": [
196,
119,
126
],
"density": 1,
"position": [
672,
511
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
825.38,
-322.28
]
},
{
"color": [
154,
97,
177
],
"density": 1,
"position": [
715,
748
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-589.51,
-767.61
]
},
{
"color": [
231,
236,
217
],
"density": 1,
"position": [
670,
664
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-94.33,
869.36
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 700,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1645567
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (854, 835), initial velocity (-220.23, -819.87) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (989, 854), initial velocity (898.46, -399.29) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (601, 531), initial velocity (488.34, -841.93) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (811, 898), initial velocity (630.46, -652.75) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (755, 530), initial velocity (822.58, 70.65) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[831.98, 753.02], [1062.56, 804.68], [649.83, 446.81], [874.04, 832.72], [837.25, 537.06]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[809.96, 671.04], [1052.21, 706.9], [698.66, 525.95], [937.07, 767.43], [919.5, 544.11]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[787.88, 588.85], [1014.35, 624.76], [747.61, 610.36], [1000.27, 701.99], [984.33, 561.37]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[765.86, 506.87], [994.0, 634.65], [796.43, 694.57], [1037.86, 663.11], [897.73, 584.55]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[743.84, 460.26], [890.04, 608.06], [845.26, 778.77], [1015.73, 723.12], [792.99, 589.58]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[721.82, 542.24], [786.09, 581.47], [894.09, 862.97], [993.59, 783.13], [688.26, 594.6]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[668.01, 624.22], [745.7, 554.89], [942.92, 947.18], [971.44, 843.14], [583.52, 599.63]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[591.35, 706.21], [751.0, 528.3], [917.92, 988.76], [949.3, 903.15], [478.99, 604.78]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[514.69, 788.19], [756.29, 501.71], [820.59, 988.56], [927.15, 963.16], [527.01, 698.02]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[477.57, 847.35], [761.59, 475.12], [723.25, 988.37], [905.01, 1023.17], [575.03, 791.26]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[586.9, 821.96], [766.89, 448.54], [625.91, 988.17], [882.87, 1031.68], [623.05, 884.5]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[696.22, 796.56], [772.19, 463.19], [538.78, 982.08], [860.72, 971.67], [671.07, 977.73]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[805.28, 771.24], [777.47, 489.71], [587.14, 897.89], [838.64, 911.81], [718.98, 1038.77]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[914.6, 745.85], [782.77, 516.3], [635.63, 813.49], [816.49, 851.8], [767.0, 945.54]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[1023.93, 720.46], [788.07, 542.88], [684.11, 729.09], [794.35, 791.79], [815.02, 852.3]]) <= 50.0"
] | ballsim | G_3_957 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
174,
96,
85
],
"density": 1,
"position": [
854,
835
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-220.23,
-819.87
]
},
{
"color": [
132,
116,
152
],
"density": 1,
"position": [
989,
854
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
898.46,
-399.29
]
},
{
"color": [
110,
233,
147
],
"density": 1,
"position": [
601,
531
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
488.34,
-841.93
]
},
{
"color": [
227,
129,
238
],
"density": 1,
"position": [
811,
898
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
630.46,
-652.75
]
},
{
"color": [
133,
234,
153
],
"density": 1,
"position": [
755,
530
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
822.58,
70.65
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 957,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2341497
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (813, 834), initial velocity (-160.71, 922.24) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (593, 707), initial velocity (699.01, -594.33) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (758, 701), initial velocity (-973.31, 134.11) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (647, 913), initial velocity (-820.04, 78.97) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (810, 953), initial velocity (21.9, -910.87) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (742, 547), initial velocity (208.79, 946.89) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[769.7, 842.42], [662.9, 647.58], [660.66, 714.4], [573.84, 912.04], [841.95, 953.48], [762.87, 641.68]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[703.81, 781.31], [732.79, 588.15], [563.32, 727.81], [581.72, 830.04], [819.84, 951.19], [783.75, 736.36]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[637.75, 720.05], [802.87, 528.58], [465.75, 741.24], [589.63, 747.82], [743.18, 894.41], [804.67, 831.27]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[611.44, 630.19], [818.94, 522.98], [421.61, 745.35], [554.29, 697.24], [666.72, 837.77], [825.55, 925.95]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[623.68, 551.13], [759.51, 592.88], [518.94, 731.95], [501.38, 659.42], [590.25, 781.13], [829.16, 1003.36]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[725.22, 561.36], [700.09, 662.77], [595.72, 655.3], [531.35, 704.48], [531.08, 777.69], [734.48, 982.49]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[826.75, 571.59], [640.67, 732.67], [661.22, 543.95], [569.16, 757.38], [481.39, 803.45], [639.8, 961.61]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[928.29, 581.82], [623.7, 771.72], [726.73, 432.6], [564.52, 841.14], [482.16, 778.75], [593.69, 892.18]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[939.71, 682.16], [636.0, 789.49], [707.77, 468.79], [546.56, 805.3], [507.91, 729.07], [645.18, 850.06]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[949.94, 783.7], [648.31, 807.26], [642.27, 580.14], [588.79, 684.93], [533.67, 679.39], [710.76, 818.18]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[960.17, 885.24], [660.61, 825.04], [699.72, 568.54], [595.62, 668.0], [523.24, 661.56], [776.34, 786.29]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[860.89, 877.27], [672.92, 842.81], [818.56, 495.54], [621.37, 702.72], [572.93, 774.42], [841.91, 754.41]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[759.6, 867.06], [685.19, 860.54], [777.71, 582.13], [647.07, 737.35], [622.49, 887.01], [907.32, 722.6]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[706.11, 829.08], [645.0, 908.63], [704.71, 700.98], [672.82, 772.06], [672.17, 999.87], [972.9, 690.72]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[729.13, 818.07], [623.51, 927.86], [602.65, 730.41], [663.41, 838.7], [765.69, 1068.92], [1008.62, 688.69]]) <= 50.0"
] | ballsim | G_3_940 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
145,
94,
62
],
"density": 1,
"position": [
813,
834
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-160.71,
922.24
]
},
{
"color": [
203,
65,
51
],
"density": 1,
"position": [
593,
707
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
699.01,
-594.33
]
},
{
"color": [
111,
183,
99
],
"density": 1,
"position": [
758,
701
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-973.31,
134.11
]
},
{
"color": [
205,
136,
217
],
"density": 1,
"position": [
647,
913
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-820.04,
78.97
]
},
{
"color": [
136,
55,
177
],
"density": 1,
"position": [
810,
953
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
21.9,
-910.87
]
},
{
"color": [
205,
113,
167
],
"density": 1,
"position": [
742,
547
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
208.79,
946.89
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 940,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3939365
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (670, 837), initial velocity (-795.65, -396.19) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (747, 712), initial velocity (-103.92, -951.79) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (759, 597), initial velocity (790.9, -579.53) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (609, 785), initial velocity (834.37, 120.21) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (854, 852), initial velocity (299.07, 928.03) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (859, 768), initial velocity (835.98, 539.17) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[611.25, 861.46], [736.6, 616.81], [788.84, 624.34], [671.62, 732.95], [816.19, 827.5], [942.59, 821.91]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[594.96, 916.46], [710.49, 548.84], [778.2, 721.8], [724.3, 650.35], [750.78, 755.21], [946.63, 737.99]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[643.91, 971.6], [622.65, 587.53], [767.53, 819.51], [773.48, 573.85], [685.21, 682.74], [916.69, 681.75]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[649.75, 952.15], [589.89, 586.03], [756.88, 916.97], [728.29, 660.79], [653.53, 644.18], [828.19, 727.19]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[626.53, 882.38], [627.87, 486.44], [696.61, 928.45], [683.1, 747.73], [708.71, 692.47], [739.69, 772.62]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[582.0, 832.89], [600.01, 587.08], [606.89, 888.93], [637.4, 797.95], [763.89, 740.76], [677.23, 833.08]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[664.81, 808.08], [662.88, 687.71], [627.85, 849.4], [647.75, 773.16], [819.06, 789.05], [682.74, 939.62]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[741.76, 800.97], [707.83, 733.21], [698.91, 842.23], [653.14, 733.17], [874.24, 837.34], [625.87, 904.93]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[853.98, 817.47], [725.97, 696.2], [721.59, 918.86], [658.54, 693.18], [903.88, 817.67], [632.92, 806.32]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[839.75, 833.97], [744.11, 659.19], [712.02, 939.64], [663.94, 653.19], [1029.81, 745.75], [726.7, 707.71]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[825.52, 850.48], [762.25, 565.2], [656.99, 881.68], [669.33, 613.2], [984.94, 727.54], [820.47, 652.95]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[811.28, 866.98], [682.98, 627.28], [601.96, 823.72], [674.73, 573.21], [878.67, 712.83], [859.22, 619.73]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[797.09, 883.45], [599.91, 696.13], [597.97, 765.91], [680.11, 533.32], [821.13, 847.28], [757.63, 616.21]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[686.83, 912.41], [609.04, 725.79], [713.57, 768.53], [663.39, 531.64], [800.61, 867.68], [655.73, 612.79]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[615.13, 890.29], [631.18, 733.68], [862.69, 804.69], [631.45, 556.33], [781.84, 882.72], [611.19, 609.37]]) <= 50.0"
] | ballsim | G_3_800 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
91,
159,
205
],
"density": 1,
"position": [
670,
837
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-795.65,
-396.19
]
},
{
"color": [
224,
109,
62
],
"density": 1,
"position": [
747,
712
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-103.92,
-951.79
]
},
{
"color": [
199,
128,
162
],
"density": 1,
"position": [
759,
597
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
790.9,
-579.53
]
},
{
"color": [
168,
207,
203
],
"density": 1,
"position": [
609,
785
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
834.37,
120.21
]
},
{
"color": [
125,
105,
201
],
"density": 1,
"position": [
854,
852
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
299.07,
928.03
]
},
{
"color": [
88,
180,
116
],
"density": 1,
"position": [
859,
768
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
835.98,
539.17
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 800,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6597603
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (699, 806), initial velocity (-572.58, 728.85) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (779, 757), initial velocity (-188.21, 823.29) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (900, 834), initial velocity (-890.4, 45.39) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (918, 588), initial velocity (-789.99, 561.73) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (462, 688), initial velocity (737.49, 445.95) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (811, 835), initial velocity (904.91, 215.82) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[641.75, 878.88], [760.18, 839.32], [901.06, 786.52], [839.0, 644.18], [535.75, 732.6], [811.4, 908.6]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[584.5, 951.75], [741.35, 921.65], [939.29, 717.57], [759.99, 700.35], [609.51, 777.21], [774.64, 1003.67]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[566.96, 1001.8], [710.99, 992.69], [977.62, 648.46], [680.79, 756.67], [683.45, 821.92], [746.62, 1024.73]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[612.25, 1035.11], [628.25, 918.18], [1001.91, 587.56], [601.79, 812.85], [757.2, 866.53], [794.82, 1014.64]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[600.98, 984.5], [602.07, 787.1], [923.1, 586.19], [522.78, 869.02], [830.96, 911.13], [843.01, 1004.56]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[589.7, 894.63], [575.9, 656.02], [844.29, 584.83], [522.35, 879.83], [904.71, 955.74], [891.2, 994.47]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[656.52, 849.86], [549.73, 524.94], [765.48, 583.46], [550.37, 804.8], [978.47, 949.74], [939.4, 1034.99]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[753.76, 822.63], [639.15, 460.61], [686.68, 582.09], [554.99, 716.25], [967.14, 890.52], [878.26, 1016.47]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[851.01, 795.41], [765.73, 472.76], [607.87, 580.72], [559.6, 627.7], [939.01, 821.6], [815.54, 997.03]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[875.08, 725.94], [892.32, 515.63], [602.85, 536.76], [493.02, 583.24], [967.2, 785.2], [752.82, 977.6]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[845.66, 625.57], [982.19, 579.71], [624.36, 477.48], [567.4, 650.82], [1036.57, 772.58], [690.1, 958.17]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[816.24, 525.21], [956.02, 710.79], [645.86, 466.95], [641.79, 718.4], [1095.89, 754.15], [627.38, 938.73]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[786.9, 465.4], [929.91, 841.54], [667.32, 526.09], [716.0, 785.81], [1067.62, 690.58], [564.82, 919.35]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[757.48, 565.76], [903.74, 972.62], [688.83, 585.37], [790.39, 853.38], [1022.02, 636.82], [502.1, 899.91]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[736.45, 670.96], [877.57, 1005.81], [703.89, 640.92], [864.78, 920.96], [976.41, 583.06], [485.83, 853.66]]) <= 50.0"
] | ballsim | G_3_792 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
65,
242,
216
],
"density": 1,
"position": [
699,
806
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-572.58,
728.85
]
},
{
"color": [
143,
214,
122
],
"density": 1,
"position": [
779,
757
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-188.21,
823.29
]
},
{
"color": [
105,
217,
169
],
"density": 1,
"position": [
900,
834
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-890.4,
45.39
]
},
{
"color": [
175,
166,
254
],
"density": 1,
"position": [
918,
588
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-789.99,
561.73
]
},
{
"color": [
147,
83,
160
],
"density": 1,
"position": [
462,
688
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
737.49,
445.95
]
},
{
"color": [
51,
73,
79
],
"density": 1,
"position": [
811,
835
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
904.91,
215.82
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 792,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 829267
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (539, 701), initial velocity (-31.31, -833.47) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (598, 772), initial velocity (879.12, 98.54) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (906, 676), initial velocity (398.45, 887.53) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (977, 815), initial velocity (631.6, -617.28) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (720, 750), initial velocity (-39.47, 874.94) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[535.88, 617.65], [685.92, 781.86], [945.84, 764.75], [1040.17, 753.28], [716.04, 837.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[532.75, 534.3], [773.83, 791.73], [985.69, 853.49], [1049.92, 730.37], [712.09, 925.0]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[572.2, 581.8], [861.96, 801.61], [946.46, 884.94], [971.5, 771.47], [708.12, 1012.72]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[618.66, 651.06], [892.55, 868.79], [849.74, 874.47], [941.5, 764.23], [743.38, 979.58]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[665.12, 720.33], [873.31, 985.78], [753.01, 864.0], [953.41, 715.09], [791.62, 906.48]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[711.58, 789.59], [865.82, 1066.64], [656.28, 853.52], [965.33, 665.94], [839.86, 833.39]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[758.04, 858.85], [919.02, 960.68], [559.55, 843.05], [977.24, 616.8], [888.11, 760.29]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[804.5, 928.11], [972.21, 854.72], [462.83, 832.58], [967.36, 583.49], [936.35, 687.2]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[850.96, 997.38], [1025.41, 748.77], [547.15, 822.1], [916.95, 579.63], [984.59, 614.1]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[864.6, 1042.81], [1011.18, 691.81], [643.88, 811.63], [866.53, 575.77], [925.14, 619.25]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[784.37, 1020.05], [893.97, 709.68], [740.61, 801.15], [816.12, 571.92], [840.71, 642.54]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[704.15, 997.27], [776.76, 727.55], [837.34, 790.68], [765.7, 568.06], [756.29, 665.83]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[624.12, 974.55], [659.84, 745.38], [933.82, 780.23], [715.41, 564.21], [672.08, 689.06]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[543.9, 951.77], [542.63, 763.25], [1030.55, 769.76], [665.0, 560.35], [587.65, 712.36]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[469.59, 929.0], [507.85, 781.12], [1089.27, 731.67], [614.58, 556.5], [503.23, 735.65]]) <= 50.0"
] | ballsim | G_3_325 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
169,
145,
124
],
"density": 1,
"position": [
539,
701
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-31.31,
-833.47
]
},
{
"color": [
90,
234,
135
],
"density": 1,
"position": [
598,
772
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
879.12,
98.54
]
},
{
"color": [
149,
250,
224
],
"density": 1,
"position": [
906,
676
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
398.45,
887.53
]
},
{
"color": [
127,
100,
252
],
"density": 1,
"position": [
977,
815
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
631.6,
-617.28
]
},
{
"color": [
205,
110,
65
],
"density": 1,
"position": [
720,
750
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-39.47,
874.94
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 325,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 674320
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (853, 634), initial velocity (833.28, 335.24) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (752, 890), initial velocity (-78.07, 989.25) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (738, 622), initial velocity (-57.55, -928.12) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (529, 809), initial velocity (561.72, -611.17) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (958, 732), initial velocity (603.92, 770.01) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (602, 762), initial velocity (490.26, 864.21) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[936.33, 667.52], [744.19, 988.93], [732.24, 529.18], [585.18, 747.89], [1018.4, 809.0], [651.02, 848.43]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[1019.65, 701.04], [760.07, 1014.94], [737.83, 471.29], [641.35, 686.78], [980.35, 814.48], [700.05, 934.85]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[1087.56, 746.01], [812.03, 930.11], [787.85, 549.95], [697.67, 625.52], [888.23, 780.75], [749.19, 1021.49]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1023.67, 784.73], [863.86, 845.49], [837.76, 628.42], [753.85, 564.41], [796.33, 747.11], [821.89, 1035.04]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[936.56, 806.58], [915.69, 760.87], [887.66, 706.89], [810.02, 503.3], [704.44, 713.47], [912.34, 993.95]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[849.45, 828.43], [1029.08, 720.96], [876.01, 740.64], [866.2, 442.18], [612.54, 679.83], [943.64, 909.87]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[762.34, 850.29], [1033.48, 781.65], [849.61, 763.69], [810.54, 462.31], [520.65, 646.18], [954.77, 811.14]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[675.23, 872.14], [966.13, 894.47], [823.22, 786.73], [735.05, 496.85], [504.5, 612.54], [965.91, 712.41]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[588.12, 893.99], [898.77, 1007.29], [796.83, 809.78], [659.56, 531.39], [596.4, 578.9], [977.04, 613.68]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[501.01, 915.84], [855.92, 1044.69], [770.44, 832.83], [740.23, 475.77], [610.21, 590.34], [917.54, 566.27]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[519.36, 937.69], [867.74, 913.83], [744.05, 855.88], [868.24, 440.64], [598.74, 616.39], [820.2, 546.34]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[606.47, 959.54], [879.55, 782.97], [717.66, 878.92], [811.92, 586.42], [587.26, 642.44], [722.86, 526.42]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[693.36, 981.33], [891.34, 652.44], [691.33, 901.91], [755.73, 731.83], [575.82, 668.43], [625.76, 506.55]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[780.47, 1003.18], [903.16, 521.58], [664.94, 924.96], [699.41, 877.61], [564.34, 694.48], [548.05, 547.04]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[867.58, 1025.04], [821.99, 458.26], [595.62, 979.19], [721.67, 966.28], [552.87, 720.53], [481.01, 620.38]]) <= 50.0"
] | ballsim | G_3_580 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
243,
127,
139
],
"density": 1,
"position": [
853,
634
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
833.28,
335.24
]
},
{
"color": [
78,
232,
66
],
"density": 1,
"position": [
752,
890
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-78.07,
989.25
]
},
{
"color": [
146,
241,
228
],
"density": 1,
"position": [
738,
622
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-57.55,
-928.12
]
},
{
"color": [
90,
246,
230
],
"density": 1,
"position": [
529,
809
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
561.72,
-611.17
]
},
{
"color": [
225,
222,
135
],
"density": 1,
"position": [
958,
732
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
603.92,
770.01
]
},
{
"color": [
194,
226,
122
],
"density": 1,
"position": [
602,
762
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
490.26,
864.21
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 580,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5328456
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (849, 557), initial velocity (772.81, 511.75) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (1024, 645), initial velocity (-868.41, 80.56) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (792, 894), initial velocity (133.36, 876.17) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (649, 555), initial velocity (-133.34, -827.46) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (731, 703), initial velocity (850.27, -46.83) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[926.27, 608.17], [937.16, 653.06], [805.33, 981.62], [635.67, 472.25], [816.03, 698.31]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[1003.54, 659.34], [850.32, 661.11], [818.66, 1045.63], [622.34, 499.04], [901.07, 693.62]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[1081.02, 710.64], [763.26, 669.19], [832.02, 957.79], [608.98, 581.99], [986.32, 688.93]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1068.11, 790.27], [676.42, 677.25], [845.35, 870.17], [595.65, 664.73], [1064.86, 687.98]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[985.15, 831.6], [589.58, 685.3], [858.68, 782.55], [582.32, 747.47], [1018.28, 759.26]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[902.19, 872.93], [502.74, 693.36], [872.01, 694.92], [568.99, 830.2], [971.73, 830.55]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[819.23, 914.27], [426.64, 707.62], [885.34, 607.3], [555.66, 912.94], [925.17, 901.84]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[736.27, 955.6], [463.08, 786.84], [898.67, 519.68], [542.33, 995.68], [878.61, 973.13]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[653.31, 996.93], [499.52, 866.07], [912.0, 453.09], [616.89, 1027.68], [832.05, 1044.42]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[672.21, 964.27], [535.96, 945.29], [925.33, 540.71], [593.36, 979.95], [785.5, 999.15]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[700.28, 924.94], [572.42, 1024.51], [938.66, 628.33], [560.65, 869.45], [738.94, 927.86]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[655.69, 843.65], [608.87, 1011.1], [951.99, 715.95], [527.93, 758.95], [758.87, 894.96]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[610.96, 762.4], [645.22, 932.07], [965.29, 803.36], [495.3, 648.73], [778.98, 862.28]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[566.11, 680.96], [681.67, 852.85], [978.62, 890.98], [549.08, 588.16], [799.15, 829.52]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[521.26, 599.51], [718.12, 773.63], [963.53, 962.19], [661.14, 561.23], [819.31, 796.75]]) <= 50.0"
] | ballsim | G_3_967 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
165,
212,
126
],
"density": 1,
"position": [
849,
557
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
772.81,
511.75
]
},
{
"color": [
221,
229,
163
],
"density": 1,
"position": [
1024,
645
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-868.41,
80.56
]
},
{
"color": [
70,
231,
84
],
"density": 1,
"position": [
792,
894
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
133.36,
876.17
]
},
{
"color": [
180,
113,
116
],
"density": 1,
"position": [
649,
555
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-133.34,
-827.46
]
},
{
"color": [
150,
95,
55
],
"density": 1,
"position": [
731,
703
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
850.27,
-46.83
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 967,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4707379
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (723, 891), initial velocity (-949.67, -265.01) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (745, 792), initial velocity (-275.86, 865.75) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (807, 839), initial velocity (-633.01, 634.02) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (851, 658), initial velocity (914.39, -165.33) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (767, 751), initial velocity (-517.18, 852.43) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[628.03, 864.51], [717.41, 878.57], [743.69, 902.4], [919.72, 680.81], [715.29, 836.25]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[631.96, 838.02], [659.43, 934.75], [699.75, 970.84], [951.11, 768.27], [647.71, 905.63]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[727.17, 811.47], [599.3, 989.06], [640.99, 974.66], [942.04, 785.7], [646.91, 913.34]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[822.14, 784.98], [603.51, 1005.92], [580.59, 975.37], [882.01, 714.78], [776.6, 920.61]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[917.11, 758.49], [655.94, 990.33], [586.64, 909.64], [821.97, 643.86], [844.26, 820.46]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[1012.08, 732.0], [684.22, 928.08], [598.18, 838.4], [761.94, 572.94], [902.8, 704.51]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[1035.41, 753.15], [712.49, 865.82], [609.73, 767.16], [678.69, 542.24], [886.04, 719.02]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[964.98, 684.15], [740.76, 803.56], [621.28, 695.92], [587.25, 525.71], [814.9, 827.71]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[872.99, 652.51], [769.03, 741.31], [632.83, 624.68], [668.54, 510.33], [743.75, 936.4]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[778.02, 626.02], [797.3, 679.05], [644.38, 553.43], [699.93, 597.78], [625.92, 964.21]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[712.16, 582.72], [825.57, 616.8], [638.96, 511.57], [702.22, 702.04], [668.78, 971.49]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[662.62, 529.98], [802.81, 642.9], [583.03, 557.2], [688.18, 815.74], [746.29, 888.39]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[613.0, 477.74], [763.11, 698.36], [630.15, 602.72], [674.18, 929.15], [804.69, 772.74]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[603.62, 493.83], [723.32, 753.95], [686.08, 648.35], [618.95, 971.49], [863.24, 656.79]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[624.52, 563.1], [683.52, 809.54], [742.02, 693.98], [651.54, 926.79], [824.93, 708.61]]) <= 50.0"
] | ballsim | G_3_576 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
134,
250,
136
],
"density": 1,
"position": [
723,
891
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-949.67,
-265.01
]
},
{
"color": [
224,
179,
183
],
"density": 1,
"position": [
745,
792
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-275.86,
865.75
]
},
{
"color": [
87,
70,
50
],
"density": 1,
"position": [
807,
839
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-633.01,
634.02
]
},
{
"color": [
250,
155,
176
],
"density": 1,
"position": [
851,
658
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
914.39,
-165.33
]
},
{
"color": [
255,
136,
62
],
"density": 1,
"position": [
767,
751
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-517.18,
852.43
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 576,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3046610
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (939, 680), initial velocity (-53.71, -886.98) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (682, 705), initial velocity (446.96, -715.58) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (838, 847), initial velocity (828.51, 324.31) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (827, 745), initial velocity (827.07, -241.51) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (799, 555), initial velocity (-819.19, -490.2) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[933.63, 591.3], [726.7, 633.44], [920.86, 879.42], [909.71, 720.85], [717.09, 505.98]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[928.26, 502.61], [771.4, 561.88], [1003.72, 911.84], [992.43, 696.71], [635.18, 456.95]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[922.87, 471.45], [816.22, 490.15], [945.78, 862.94], [1052.69, 685.58], [553.07, 477.33]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[917.5, 560.15], [860.92, 466.55], [876.27, 807.4], [990.43, 745.13], [550.63, 572.25]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[912.13, 648.84], [905.62, 538.11], [806.77, 751.86], [928.15, 804.67], [549.14, 667.7]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[906.76, 737.54], [950.32, 609.67], [737.26, 696.31], [865.86, 864.22], [547.65, 763.16]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[901.39, 826.24], [995.03, 681.22], [667.75, 640.77], [803.58, 923.77], [546.16, 858.62]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[896.02, 914.93], [1039.72, 752.78], [598.25, 585.23], [741.3, 983.31], [544.67, 954.08]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[890.65, 1003.63], [1050.9, 805.0], [528.74, 529.69], [679.02, 1042.87], [580.34, 1028.09]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[885.28, 1022.51], [966.57, 802.07], [601.8, 556.45], [708.91, 1049.22], [571.6, 1021.31]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[879.9, 933.82], [882.25, 799.14], [684.67, 588.88], [801.8, 1019.19], [540.19, 954.89]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[874.53, 845.12], [797.92, 796.21], [767.53, 621.3], [894.7, 989.16], [538.7, 871.2]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[869.18, 756.65], [713.81, 793.29], [850.18, 653.64], [982.62, 956.47], [537.22, 787.71]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[863.8, 667.95], [629.48, 790.36], [933.04, 686.06], [962.19, 861.02], [535.73, 704.02]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[858.43, 579.25], [545.15, 787.43], [1015.9, 718.48], [941.75, 765.56], [534.24, 620.33]]) <= 50.0"
] | ballsim | G_3_988 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
55,
95,
173
],
"density": 1,
"position": [
939,
680
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-53.71,
-886.98
]
},
{
"color": [
247,
54,
245
],
"density": 1,
"position": [
682,
705
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
446.96,
-715.58
]
},
{
"color": [
220,
91,
164
],
"density": 1,
"position": [
838,
847
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
828.51,
324.31
]
},
{
"color": [
92,
106,
66
],
"density": 1,
"position": [
827,
745
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
827.07,
-241.51
]
},
{
"color": [
221,
69,
161
],
"density": 1,
"position": [
799,
555
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-819.19,
-490.2
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 988,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4524572
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (849, 556), initial velocity (-794.31, -399.82) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (812, 775), initial velocity (26.02, 915.67) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (963, 803), initial velocity (-127.62, 931.1) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (537, 883), initial velocity (591.43, 759.0) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (847, 820), initial velocity (291.45, 924.71) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[769.58, 516.01], [814.61, 866.58], [950.23, 896.12], [596.13, 958.9], [876.15, 912.48]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[690.16, 476.03], [817.22, 958.15], [937.46, 989.23], [655.26, 1034.8], [905.3, 1004.96]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[610.54, 454.54], [819.84, 1049.95], [947.2, 979.97], [714.54, 1000.58], [870.38, 1039.92]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[548.2, 504.38], [774.62, 1056.16], [994.68, 882.51], [773.67, 924.67], [814.73, 961.28]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[553.28, 593.16], [717.34, 1046.59], [1042.16, 785.05], [821.82, 814.93], [776.64, 905.08]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[558.36, 681.92], [660.07, 1034.48], [1058.62, 705.49], [866.47, 694.42], [742.04, 859.65]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[563.44, 770.69], [602.79, 1022.37], [950.46, 697.87], [911.12, 573.91], [707.45, 814.21]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[568.52, 859.46], [545.52, 1010.24], [842.31, 690.25], [922.74, 472.48], [672.85, 768.78]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[580.09, 936.99], [553.28, 972.11], [734.15, 682.63], [796.05, 450.9], [638.26, 723.35]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[637.0, 936.01], [610.53, 985.99], [635.93, 675.02], [669.37, 459.22], [598.24, 677.91]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[693.91, 935.03], [712.82, 973.86], [622.91, 667.4], [558.75, 490.08], [511.67, 632.48]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[750.82, 934.06], [815.12, 961.72], [609.9, 659.78], [603.41, 610.59], [525.55, 645.03]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[807.59, 933.08], [917.16, 949.62], [597.57, 787.31], [667.28, 671.27], [588.3, 682.84]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[864.49, 932.11], [985.39, 917.82], [617.27, 961.62], [734.64, 721.86], [630.36, 705.59]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[921.4, 931.13], [944.74, 823.17], [636.97, 978.92], [802.0, 772.44], [672.43, 728.34]]) <= 50.0"
] | ballsim | G_3_841 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
55,
64,
189
],
"density": 1,
"position": [
849,
556
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-794.31,
-399.82
]
},
{
"color": [
135,
211,
230
],
"density": 1,
"position": [
812,
775
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
26.02,
915.67
]
},
{
"color": [
138,
83,
59
],
"density": 1,
"position": [
963,
803
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-127.62,
931.1
]
},
{
"color": [
100,
210,
115
],
"density": 1,
"position": [
537,
883
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
591.43,
759
]
},
{
"color": [
193,
110,
96
],
"density": 1,
"position": [
847,
820
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
291.45,
924.71
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 841,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 9151730
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (955, 676), initial velocity (552.59, -580.48) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (804, 736), initial velocity (-833.21, 380.29) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (866, 713), initial velocity (4.83, -901.88) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (808, 922), initial velocity (641.09, 508.39) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (767, 701), initial velocity (781.9, -364.53) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[972.93, 655.26], [720.67, 774.04], [866.49, 622.81], [872.11, 972.83], [845.2, 664.55]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[914.87, 710.51], [637.35, 812.07], [866.98, 532.63], [831.33, 918.76], [923.4, 628.1]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[856.67, 765.9], [553.82, 850.21], [797.19, 512.49], [780.37, 854.49], [941.34, 652.01]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[798.61, 821.15], [540.58, 818.16], [707.01, 512.98], [729.54, 790.38], [904.89, 730.2]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[740.56, 876.4], [578.61, 734.83], [631.55, 528.19], [678.71, 726.27], [868.44, 808.4]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[682.5, 931.65], [597.87, 651.51], [631.06, 618.38], [646.66, 662.16], [831.99, 886.6]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[631.91, 979.43], [525.44, 664.01], [627.86, 629.58], [747.58, 660.93], [795.54, 964.8]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[687.16, 921.37], [534.11, 789.5], [630.28, 625.67], [855.81, 667.01], [759.09, 1042.99]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[742.41, 863.32], [567.32, 890.44], [632.7, 621.76], [964.04, 673.09], [774.13, 1069.68]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[797.66, 805.26], [692.81, 899.11], [635.12, 617.86], [1034.16, 717.28], [816.56, 997.47]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[852.91, 747.2], [817.67, 906.69], [637.53, 613.95], [1029.48, 814.72], [853.35, 919.87]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[908.15, 689.15], [865.74, 781.23], [639.95, 610.04], [921.26, 808.64], [932.11, 914.96]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[963.27, 631.24], [913.7, 656.08], [642.36, 606.15], [813.3, 802.58], [939.72, 839.11]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[928.15, 663.54], [885.6, 606.78], [644.78, 602.24], [705.07, 796.5], [944.63, 760.35]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[870.1, 718.79], [760.14, 654.85], [647.19, 598.34], [596.84, 790.42], [949.54, 681.59]]) <= 50.0"
] | ballsim | G_3_370 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
65,
134,
230
],
"density": 1,
"position": [
955,
676
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
552.59,
-580.48
]
},
{
"color": [
220,
217,
71
],
"density": 1,
"position": [
804,
736
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-833.21,
380.29
]
},
{
"color": [
80,
138,
76
],
"density": 1,
"position": [
866,
713
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
4.83,
-901.88
]
},
{
"color": [
159,
247,
231
],
"density": 1,
"position": [
808,
922
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
641.09,
508.39
]
},
{
"color": [
219,
64,
125
],
"density": 1,
"position": [
767,
701
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
781.9,
-364.53
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 370,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 9112988
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (619, 953), initial velocity (-472.82, -835.41) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (558, 693), initial velocity (-516.37, -746.22) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (849, 839), initial velocity (-737.19, -446.43) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (975, 697), initial velocity (629.59, 703.04) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (742, 752), initial velocity (-784.6, -280.6) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (676, 606), initial velocity (64.04, 797.63) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[571.71, 869.46], [506.36, 618.37], [775.27, 794.35], [1037.96, 767.31], [663.53, 723.95], [682.4, 685.76]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[524.42, 785.91], [543.78, 595.15], [725.29, 763.41], [1031.14, 797.35], [585.07, 695.9], [657.94, 747.71]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[477.02, 702.16], [634.46, 602.56], [750.77, 776.04], [938.55, 777.94], [506.4, 667.77], [535.25, 753.15]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[476.9, 708.44], [724.92, 609.96], [776.18, 788.64], [846.19, 758.58], [485.23, 624.61], [412.87, 758.57]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[554.97, 774.51], [815.37, 617.36], [801.6, 801.23], [753.83, 739.22], [562.56, 627.03], [459.37, 666.48]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[633.05, 840.57], [905.83, 624.76], [827.01, 813.83], [661.47, 719.86], [639.88, 629.44], [525.24, 563.21]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[711.13, 906.63], [996.28, 632.15], [852.43, 826.43], [569.11, 700.5], [717.2, 631.86], [591.11, 459.93]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[789.2, 972.7], [1018.47, 678.96], [877.84, 839.03], [476.75, 681.14], [794.52, 634.28], [656.98, 533.85]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[867.28, 1038.76], [979.68, 760.99], [903.26, 851.62], [481.19, 717.66], [871.83, 636.69], [722.85, 637.13]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[945.36, 1004.67], [956.45, 834.04], [920.89, 868.71], [544.15, 787.98], [949.15, 639.11], [788.72, 740.4]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[982.44, 914.94], [1021.95, 855.84], [894.16, 911.41], [607.11, 858.29], [1026.47, 641.53], [854.59, 843.67]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[979.03, 814.31], [1036.46, 829.0], [867.43, 980.36], [670.08, 928.6], [1013.73, 695.96], [920.45, 912.87]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[913.53, 730.55], [1069.38, 783.29], [840.76, 1039.33], [732.88, 998.74], [999.42, 751.18], [986.16, 947.62]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[794.31, 677.5], [1095.43, 741.48], [814.03, 943.92], [795.85, 1042.41], [1026.3, 782.75], [930.14, 912.09]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[675.1, 624.45], [1039.23, 747.17], [787.29, 848.51], [858.81, 972.09], [1053.16, 814.31], [867.03, 872.47]]) <= 50.0"
] | ballsim | G_3_28 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
194,
116,
183
],
"density": 1,
"position": [
619,
953
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-472.82,
-835.41
]
},
{
"color": [
71,
77,
92
],
"density": 1,
"position": [
558,
693
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-516.37,
-746.22
]
},
{
"color": [
203,
83,
132
],
"density": 1,
"position": [
849,
839
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-737.19,
-446.43
]
},
{
"color": [
204,
181,
246
],
"density": 1,
"position": [
975,
697
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
629.59,
703.04
]
},
{
"color": [
61,
94,
97
],
"density": 1,
"position": [
742,
752
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-784.6,
-280.6
]
},
{
"color": [
186,
213,
142
],
"density": 1,
"position": [
676,
606
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
64.04,
797.63
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 28,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 525826
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (880, 1017), initial velocity (-442.04, -714.42) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (630, 685), initial velocity (-918.36, -76.03) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (784, 841), initial velocity (-836.46, -137.94) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (661, 587), initial velocity (-22.82, -958.46) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (902, 622), initial velocity (914.26, 335.31) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (749, 885), initial velocity (-467.1, -845.34) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[835.79, 945.56], [538.15, 677.41], [735.37, 792.2], [658.73, 491.15], [993.43, 655.52], [660.67, 842.08]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[791.57, 874.13], [446.32, 669.81], [700.92, 729.21], [656.46, 495.2], [1062.27, 702.08], [555.47, 816.03]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[747.25, 802.51], [491.43, 741.4], [666.39, 666.06], [654.18, 591.29], [1045.58, 798.27], [450.01, 789.92]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[703.03, 731.08], [543.93, 817.13], [666.19, 708.51], [611.18, 561.79], [1018.13, 887.98], [432.11, 713.47]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[759.95, 733.12], [596.42, 892.86], [576.08, 711.97], [554.86, 491.31], [943.37, 950.38], [483.26, 621.53]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[822.42, 739.2], [648.91, 968.59], [480.4, 711.38], [628.33, 495.76], [868.61, 1012.78], [558.41, 543.45]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[884.9, 745.27], [701.4, 1044.33], [438.56, 741.88], [717.54, 509.3], [793.86, 1034.31], [633.56, 465.38]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[947.37, 751.35], [728.13, 1020.56], [486.91, 824.45], [806.75, 522.85], [752.58, 938.43], [708.7, 503.2]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[1009.85, 757.43], [719.49, 1005.99], [535.26, 907.02], [895.96, 536.4], [757.26, 796.59], [783.85, 581.28]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[1072.34, 763.51], [710.85, 991.41], [583.6, 989.59], [985.17, 549.95], [761.95, 654.74], [859.0, 659.35]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[1073.09, 733.95], [702.2, 976.84], [631.94, 1039.3], [958.76, 630.24], [766.64, 512.9], [934.14, 737.43]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[1036.57, 682.87], [715.01, 949.88], [656.85, 970.27], [925.88, 714.27], [771.33, 519.45], [1009.29, 815.51]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[1000.15, 631.93], [775.21, 895.59], [629.89, 931.33], [893.07, 798.09], [776.0, 660.94], [991.18, 839.65]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[963.65, 580.85], [835.56, 841.17], [602.86, 892.29], [860.19, 882.13], [780.69, 802.78], [885.98, 813.6]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[927.15, 529.78], [824.9, 857.77], [575.84, 853.25], [827.3, 966.16], [785.38, 944.63], [873.03, 695.3]]) <= 50.0"
] | ballsim | G_3_727 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
198,
84,
204
],
"density": 1,
"position": [
880,
1017
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-442.04,
-714.42
]
},
{
"color": [
189,
175,
148
],
"density": 1,
"position": [
630,
685
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-918.36,
-76.03
]
},
{
"color": [
71,
65,
53
],
"density": 1,
"position": [
784,
841
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-836.46,
-137.94
]
},
{
"color": [
83,
253,
250
],
"density": 1,
"position": [
661,
587
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-22.82,
-958.46
]
},
{
"color": [
117,
177,
179
],
"density": 1,
"position": [
902,
622
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
914.26,
335.31
]
},
{
"color": [
213,
247,
148
],
"density": 1,
"position": [
749,
885
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-467.1,
-845.34
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 727,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6493945
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (660, 775), initial velocity (-474.21, -781.74) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (799, 769), initial velocity (-65.33, 981.05) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (844, 902), initial velocity (692.54, -451.42) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (571, 899), initial velocity (-655.6, 747.69) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (700, 834), initial velocity (-828.18, -207.91) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (910, 807), initial velocity (666.63, -519.61) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[612.59, 696.83], [792.46, 867.1], [913.26, 856.86], [626.73, 852.5], [617.19, 813.2], [976.67, 755.05]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[565.18, 618.65], [785.91, 965.19], [982.53, 811.72], [701.51, 786.95], [534.38, 792.4], [1043.34, 703.09]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[612.05, 634.69], [779.35, 1063.53], [1051.95, 766.46], [776.48, 721.23], [451.36, 771.55], [1000.8, 760.4]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[690.23, 682.1], [727.19, 1033.7], [1068.98, 773.53], [851.26, 655.68], [436.13, 749.25], [948.84, 827.07]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[768.4, 729.51], [733.73, 935.61], [1004.88, 823.84], [926.04, 590.13], [518.94, 770.05], [896.89, 893.75]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[846.57, 776.93], [740.27, 837.51], [935.62, 868.98], [874.36, 651.03], [601.75, 790.85], [844.94, 960.42]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[924.75, 824.34], [746.82, 739.42], [866.36, 914.12], [808.81, 725.81], [684.56, 811.65], [792.98, 1027.1]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[975.92, 844.75], [753.36, 641.32], [797.09, 959.26], [744.72, 796.07], [764.69, 840.73], [746.7, 1088.08]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[928.51, 766.58], [759.9, 543.22], [727.83, 1004.4], [700.39, 805.52], [808.66, 981.06], [813.37, 1036.13]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[881.1, 688.4], [766.45, 445.12], [701.88, 1006.24], [656.05, 814.97], [783.76, 909.41], [766.49, 948.81]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[833.68, 610.23], [727.02, 457.65], [747.03, 936.97], [611.72, 824.42], [747.97, 804.24], [703.65, 857.89]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[786.27, 532.05], [720.47, 555.75], [792.17, 867.71], [567.38, 833.87], [712.18, 699.06], [640.81, 766.98]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[738.98, 454.08], [734.87, 617.37], [837.2, 798.62], [523.16, 843.29], [634.63, 666.62], [578.12, 676.29]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[776.43, 460.78], [782.14, 622.27], [882.34, 729.36], [505.69, 825.87], [491.22, 747.82], [567.58, 637.67]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[854.61, 508.2], [829.4, 627.18], [927.48, 660.1], [515.14, 781.53], [481.36, 695.48], [658.5, 700.51]]) <= 50.0"
] | ballsim | G_3_38 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
181,
174,
238
],
"density": 1,
"position": [
660,
775
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-474.21,
-781.74
]
},
{
"color": [
65,
94,
219
],
"density": 1,
"position": [
799,
769
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-65.33,
981.05
]
},
{
"color": [
96,
197,
113
],
"density": 1,
"position": [
844,
902
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
692.54,
-451.42
]
},
{
"color": [
138,
51,
175
],
"density": 1,
"position": [
571,
899
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-655.6,
747.69
]
},
{
"color": [
69,
122,
128
],
"density": 1,
"position": [
700,
834
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-828.18,
-207.91
]
},
{
"color": [
182,
121,
239
],
"density": 1,
"position": [
910,
807
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
666.63,
-519.61
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 38,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7634147
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (720, 632), initial velocity (768.22, 331.19) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (643, 913), initial velocity (-8.99, -971.75) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (759, 774), initial velocity (734.49, -448.59) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (692, 942), initial velocity (-634.26, 570.5) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (987, 742), initial velocity (947.83, 245.42) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (772, 968), initial velocity (40.23, 833.78) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[796.83, 665.13], [642.1, 815.83], [832.44, 729.15], [652.01, 975.62], [1081.78, 766.54], [776.03, 1051.38]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[857.97, 671.09], [641.19, 718.66], [914.45, 699.15], [709.06, 912.19], [1033.42, 708.91], [719.96, 1060.57]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[890.11, 626.57], [640.29, 621.25], [938.96, 696.65], [766.26, 848.61], [1039.3, 718.19], [715.92, 976.97]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[922.16, 582.16], [639.38, 524.09], [900.71, 694.16], [823.32, 785.18], [982.35, 840.24], [711.89, 893.6]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[892.73, 599.23], [718.1, 506.53], [861.82, 689.73], [881.0, 723.69], [827.91, 864.77], [707.86, 810.23]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[848.32, 631.29], [815.27, 507.44], [814.75, 660.14], [946.85, 687.37], [673.46, 889.31], [703.84, 726.85]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[803.91, 663.34], [863.33, 557.45], [767.68, 630.55], [1000.82, 662.91], [566.18, 866.68], [699.81, 643.48]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[759.51, 695.4], [864.23, 654.62], [720.61, 600.96], [964.49, 728.75], [590.72, 712.24], [695.78, 560.1]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[715.1, 727.45], [865.13, 751.78], [673.54, 571.37], [928.17, 794.59], [615.25, 557.8], [691.75, 476.73]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[670.69, 759.51], [860.25, 843.16], [635.12, 568.4], [896.71, 865.31], [718.5, 472.25], [766.3, 460.38]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[626.28, 791.57], [823.92, 903.1], [602.7, 583.88], [887.31, 958.08], [737.76, 427.77], [780.78, 556.2]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[581.87, 823.62], [762.95, 938.41], [570.28, 599.36], [810.85, 983.81], [750.3, 414.06], [753.51, 693.78]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[537.57, 855.6], [663.98, 935.49], [538.15, 615.01], [766.68, 1041.58], [731.08, 458.43], [726.31, 831.01]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[540.0, 840.82], [577.57, 919.75], [522.67, 647.44], [752.44, 1069.44], [711.82, 502.91], [699.04, 968.58]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[568.93, 791.01], [576.67, 824.04], [507.19, 679.86], [810.35, 1025.15], [692.56, 547.4], [758.49, 1019.44]]) <= 50.0"
] | ballsim | G_3_984 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
227,
80,
152
],
"density": 1,
"position": [
720,
632
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
768.22,
331.19
]
},
{
"color": [
57,
90,
97
],
"density": 1,
"position": [
643,
913
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-8.99,
-971.75
]
},
{
"color": [
101,
150,
148
],
"density": 1,
"position": [
759,
774
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
734.49,
-448.59
]
},
{
"color": [
199,
196,
200
],
"density": 1,
"position": [
692,
942
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-634.26,
570.5
]
},
{
"color": [
61,
150,
195
],
"density": 1,
"position": [
987,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
947.83,
245.42
]
},
{
"color": [
130,
152,
246
],
"density": 1,
"position": [
772,
968
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
40.23,
833.78
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 984,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7353660
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (822, 756), initial velocity (-857.71, 178.08) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (743, 726), initial velocity (-404.71, 857.75) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (789, 831), initial velocity (-764.27, -589.21) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (663, 732), initial velocity (-824.68, 484.12) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (704, 872), initial velocity (931.05, -159.0) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (688, 656), initial velocity (946.3, 296.74) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[748.12, 761.92], [660.3, 769.54], [725.87, 813.64], [580.53, 780.41], [797.12, 856.11], [782.63, 685.66]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[701.68, 740.41], [634.72, 745.79], [673.82, 872.6], [646.96, 828.83], [890.23, 840.21], [877.26, 715.33]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[772.53, 718.85], [574.83, 721.98], [621.64, 931.7], [729.64, 877.36], [970.38, 801.42], [972.12, 745.06]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[864.89, 697.34], [678.91, 698.22], [595.43, 990.66], [806.32, 915.75], [993.59, 771.67], [1037.03, 755.45]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[957.25, 675.83], [782.99, 674.47], [609.42, 983.71], [805.64, 820.12], [893.71, 775.28], [1021.48, 750.57]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[984.86, 766.45], [887.06, 650.71], [584.37, 909.15], [743.76, 724.49], [827.26, 778.89], [938.71, 795.4]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[956.29, 760.01], [941.5, 712.93], [605.69, 834.59], [614.78, 628.86], [797.48, 782.51], [855.95, 840.22]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[929.22, 806.93], [876.56, 649.69], [630.74, 760.03], [659.22, 533.23], [767.69, 786.12], [773.18, 885.05]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[894.72, 846.79], [736.96, 677.74], [655.79, 685.47], [676.03, 631.86], [737.91, 789.73], [690.42, 929.87]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[831.19, 836.39], [626.56, 705.79], [587.15, 709.72], [809.81, 639.27], [708.12, 793.35], [607.66, 974.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[767.67, 825.99], [796.92, 733.85], [607.76, 744.41], [928.93, 683.86], [678.34, 796.96], [629.14, 1000.5]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[704.14, 815.59], [967.28, 761.9], [629.19, 770.65], [991.78, 810.07], [655.67, 807.69], [631.71, 906.4]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[703.5, 805.22], [908.13, 735.85], [603.14, 749.39], [959.02, 770.64], [637.01, 775.79], [609.73, 888.05]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[715.17, 794.82], [683.18, 739.32], [628.46, 687.6], [971.86, 810.21], [662.47, 708.82], [597.84, 932.82]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[726.84, 784.42], [626.68, 865.15], [587.56, 518.33], [958.42, 804.27], [686.32, 665.42], [640.39, 977.6]]) <= 50.0"
] | ballsim | G_3_955 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
181,
189,
165
],
"density": 1,
"position": [
822,
756
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-857.71,
178.08
]
},
{
"color": [
68,
244,
168
],
"density": 1,
"position": [
743,
726
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-404.71,
857.75
]
},
{
"color": [
166,
203,
205
],
"density": 1,
"position": [
789,
831
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-764.27,
-589.21
]
},
{
"color": [
244,
163,
212
],
"density": 1,
"position": [
663,
732
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-824.68,
484.12
]
},
{
"color": [
235,
89,
247
],
"density": 1,
"position": [
704,
872
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
931.05,
-159
]
},
{
"color": [
235,
70,
187
],
"density": 1,
"position": [
688,
656
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
946.3,
296.74
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 955,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 254069
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (730, 689), initial velocity (862.42, -200.92) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (549, 763), initial velocity (729.05, 586.01) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (769, 801), initial velocity (-680.31, -580.59) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (720, 576), initial velocity (-790.22, -485.2) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (910, 660), initial velocity (-303.31, -937.23) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (631, 532), initial velocity (-618.22, -758.5) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[816.23, 668.91], [621.9, 821.59], [700.96, 742.94], [640.97, 527.49], [879.68, 566.27], [569.18, 456.15]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[902.46, 648.81], [694.8, 880.19], [632.92, 684.89], [561.94, 478.97], [849.36, 472.55], [659.02, 467.85]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[988.91, 628.67], [767.88, 938.93], [564.7, 626.68], [627.99, 514.2], [818.96, 506.55], [755.86, 483.5]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[997.44, 653.45], [840.78, 997.52], [501.26, 571.28], [709.51, 558.37], [788.64, 600.28], [852.47, 499.11]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[936.92, 718.09], [913.68, 1053.38], [585.56, 601.16], [791.03, 602.54], [758.31, 694.0], [949.08, 514.73]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[876.39, 782.74], [952.2, 974.93], [669.86, 631.05], [872.54, 646.7], [727.99, 787.73], [944.42, 588.8]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[815.87, 847.39], [966.51, 882.5], [754.16, 660.93], [925.38, 740.56], [697.67, 881.45], [924.0, 655.42]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[755.35, 912.04], [980.82, 790.07], [838.46, 690.81], [940.83, 899.17], [667.35, 975.18], [922.27, 689.67]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[694.83, 976.69], [995.12, 697.64], [882.39, 697.38], [912.78, 1032.67], [637.02, 1045.96], [940.72, 735.58]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[634.3, 1041.34], [1009.43, 605.21], [883.17, 679.05], [767.69, 1016.27], [606.7, 952.24], [980.73, 793.93]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[573.78, 1003.52], [940.17, 561.01], [883.95, 660.71], [622.6, 950.35], [576.38, 858.52], [1020.75, 852.28]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[513.26, 938.87], [852.96, 527.17], [884.73, 642.38], [477.5, 884.43], [546.06, 764.79], [987.98, 868.62]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[473.79, 862.31], [765.97, 493.42], [885.51, 624.09], [485.6, 730.44], [515.81, 671.3], [917.6, 863.16]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[448.07, 777.57], [678.77, 459.58], [886.29, 605.75], [479.67, 608.88], [520.5, 548.38], [847.04, 857.69]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[430.85, 697.74], [616.94, 469.71], [887.07, 587.42], [564.99, 581.93], [617.3, 526.99], [776.49, 852.22]]) <= 50.0"
] | ballsim | G_3_509 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
234,
230,
79
],
"density": 1,
"position": [
730,
689
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
862.42,
-200.92
]
},
{
"color": [
193,
213,
122
],
"density": 1,
"position": [
549,
763
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
729.05,
586.01
]
},
{
"color": [
214,
63,
129
],
"density": 1,
"position": [
769,
801
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-680.31,
-580.59
]
},
{
"color": [
85,
252,
208
],
"density": 1,
"position": [
720,
576
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-790.22,
-485.2
]
},
{
"color": [
148,
199,
243
],
"density": 1,
"position": [
910,
660
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-303.31,
-937.23
]
},
{
"color": [
105,
101,
67
],
"density": 1,
"position": [
631,
532
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-618.22,
-758.5
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 509,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5044260
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (956, 757), initial velocity (314.31, 919.08) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (788, 874), initial velocity (-341.93, 752.78) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (667, 786), initial velocity (-311.21, 948.75) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (894, 973), initial velocity (508.5, -742.63) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (922, 545), initial velocity (-443.49, -854.82) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (773, 724), initial velocity (955.48, 15.29) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[987.42, 848.92], [753.8, 949.27], [635.87, 880.87], [944.85, 898.73], [877.64, 459.52], [868.56, 725.54]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[982.8, 837.76], [719.59, 1024.54], [604.74, 975.75], [940.87, 856.13], [833.28, 516.47], [964.11, 727.08]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[887.25, 743.36], [685.3, 1011.44], [573.54, 1044.0], [916.39, 825.24], [788.81, 602.15], [1059.91, 728.62]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[791.93, 649.19], [651.1, 936.17], [542.41, 949.13], [891.98, 794.43], [744.45, 687.63], [1062.57, 783.79]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[696.62, 555.03], [616.89, 860.91], [511.28, 854.26], [867.57, 763.62], [700.09, 773.1], [1016.13, 867.31]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[601.31, 460.86], [582.69, 785.64], [480.17, 759.38], [843.15, 732.81], [655.73, 858.57], [969.67, 950.83]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[535.41, 540.78], [548.49, 710.37], [449.05, 664.51], [818.74, 702.0], [611.37, 944.05], [923.21, 1034.35]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[541.71, 600.8], [536.15, 672.96], [437.88, 677.83], [794.32, 671.19], [567.01, 1029.52], [876.75, 997.01]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[613.81, 546.87], [559.61, 697.55], [465.3, 857.34], [769.91, 640.38], [647.31, 1043.06], [830.29, 913.48]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[685.9, 492.94], [583.07, 722.13], [619.52, 965.71], [745.5, 609.57], [743.52, 1047.4], [783.83, 829.96]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[758.0, 451.51], [606.53, 746.72], [773.74, 1040.78], [721.08, 578.76], [839.74, 1051.75], [737.37, 746.44]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[830.09, 505.45], [630.0, 771.3], [927.96, 932.41], [696.67, 547.95], [926.75, 1048.09], [690.91, 662.92]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[902.0, 559.25], [653.4, 795.82], [1052.23, 807.24], [672.31, 517.21], [882.5, 962.82], [644.57, 579.61]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[974.1, 613.18], [676.86, 820.41], [1009.98, 653.56], [647.9, 486.4], [838.14, 877.35], [598.11, 496.09]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[909.39, 530.31], [700.32, 844.99], [1049.63, 784.83], [633.33, 461.27], [793.78, 791.88], [589.22, 450.87]]) <= 50.0"
] | ballsim | G_3_732 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
215,
138,
114
],
"density": 1,
"position": [
956,
757
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
314.31,
919.08
]
},
{
"color": [
139,
175,
166
],
"density": 1,
"position": [
788,
874
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-341.93,
752.78
]
},
{
"color": [
81,
195,
232
],
"density": 1,
"position": [
667,
786
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-311.21,
948.75
]
},
{
"color": [
72,
190,
189
],
"density": 1,
"position": [
894,
973
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
508.5,
-742.63
]
},
{
"color": [
252,
181,
86
],
"density": 1,
"position": [
922,
545
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-443.49,
-854.82
]
},
{
"color": [
252,
134,
104
],
"density": 1,
"position": [
773,
724
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
955.48,
15.29
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 732,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2049800
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (710, 744), initial velocity (-712.86, -562.13) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (600, 791), initial velocity (-582.4, -615.45) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (636, 704), initial velocity (-425.94, -844.88) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (776, 942), initial velocity (-624.1, 502.61) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (734, 654), initial velocity (-781.11, -341.85) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[638.71, 687.8], [541.75, 729.45], [593.4, 619.5], [713.6, 992.27], [655.9, 619.82]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[567.42, 631.6], [483.5, 667.9], [550.79, 535.01], [664.52, 1001.53], [577.8, 585.64]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[518.15, 662.09], [497.45, 606.2], [555.06, 516.72], [643.52, 923.99], [524.56, 586.07]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[576.4, 759.43], [548.35, 733.53], [595.96, 526.67], [622.57, 846.65], [540.11, 527.22]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[628.6, 721.75], [526.69, 862.0], [661.9, 526.01], [644.21, 843.06], [564.7, 582.3]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[678.85, 605.3], [482.54, 925.73], [727.84, 525.35], [690.69, 882.52], [589.28, 637.38]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[729.09, 488.84], [505.3, 783.49], [793.79, 524.69], [737.18, 921.97], [613.86, 692.45]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[820.27, 498.42], [528.08, 641.25], [859.73, 524.03], [783.66, 961.42], [638.45, 747.53]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[862.73, 524.63], [561.01, 530.27], [949.5, 549.52], [830.15, 1000.88], [663.03, 802.61]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[890.95, 542.62], [663.04, 631.95], [938.22, 657.7], [876.63, 1040.33], [687.62, 857.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[919.17, 560.62], [765.06, 733.63], [926.94, 765.87], [847.53, 1024.87], [712.2, 912.77]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[947.4, 578.61], [867.09, 835.32], [915.66, 874.05], [795.63, 992.84], [736.79, 967.85]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[975.55, 596.56], [968.86, 936.75], [904.41, 981.96], [799.88, 960.89], [710.04, 1022.79]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[1003.77, 614.55], [841.19, 871.57], [843.55, 1054.15], [827.86, 928.86], [695.75, 972.59]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[1015.84, 644.29], [712.94, 805.97], [774.59, 1004.61], [855.84, 896.83], [688.89, 899.52]]) <= 50.0"
] | ballsim | G_3_405 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
156,
70,
164
],
"density": 1,
"position": [
710,
744
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-712.86,
-562.13
]
},
{
"color": [
236,
192,
235
],
"density": 1,
"position": [
600,
791
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-582.4,
-615.45
]
},
{
"color": [
248,
241,
61
],
"density": 1,
"position": [
636,
704
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-425.94,
-844.88
]
},
{
"color": [
204,
253,
124
],
"density": 1,
"position": [
776,
942
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-624.1,
502.61
]
},
{
"color": [
147,
138,
198
],
"density": 1,
"position": [
734,
654
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-781.11,
-341.85
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 405,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3894128
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (741, 760), initial velocity (305.17, -879.43) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (818, 742), initial velocity (942.59, 297.12) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (631, 772), initial velocity (922.18, -69.9) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (783, 632), initial velocity (-337.33, 926.02) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (803, 890), initial velocity (-935.52, -277.57) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[750.61, 733.16], [912.26, 771.71], [701.13, 787.1], [779.26, 672.63], [709.45, 862.24]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[839.18, 697.69], [1003.9, 796.87], [702.65, 870.81], [817.73, 632.64], [588.26, 862.11]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[913.22, 707.58], [1014.67, 718.63], [715.94, 942.96], [817.26, 649.31], [700.5, 871.88]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[987.07, 717.44], [941.82, 785.41], [680.34, 930.3], [784.45, 721.97], [831.56, 881.62]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[1060.91, 727.3], [954.92, 789.73], [624.65, 882.83], [751.63, 794.62], [829.5, 838.58]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[1088.0, 759.34], [1045.83, 733.1], [596.06, 835.37], [692.24, 867.28], [755.01, 835.11]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[996.4, 725.27], [938.76, 769.84], [651.75, 787.91], [591.16, 939.94], [722.2, 831.65]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[895.03, 733.11], [821.79, 806.58], [707.44, 740.45], [649.16, 981.19], [689.39, 828.18]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[793.67, 740.95], [704.82, 843.33], [763.12, 692.99], [636.79, 857.34], [656.57, 824.71]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[692.3, 748.79], [599.11, 888.24], [818.81, 645.53], [579.5, 840.6], [660.8, 707.26]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[590.93, 756.62], [658.69, 942.57], [850.85, 639.05], [649.59, 816.87], [665.31, 588.9]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[655.46, 764.46], [707.08, 920.02], [837.59, 711.02], [719.69, 793.14], [635.82, 529.44]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[752.49, 772.28], [706.42, 812.8], [824.36, 782.81], [791.83, 769.47], [621.55, 592.37]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[813.4, 780.11], [705.76, 705.3], [816.05, 859.73], [879.88, 741.58], [721.8, 655.46]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[874.31, 787.95], [705.1, 597.81], [787.27, 900.94], [967.62, 713.38], [822.04, 718.54]]) <= 50.0"
] | ballsim | G_3_756 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
107,
221,
255
],
"density": 1,
"position": [
741,
760
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
305.17,
-879.43
]
},
{
"color": [
131,
198,
75
],
"density": 1,
"position": [
818,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
942.59,
297.12
]
},
{
"color": [
140,
206,
138
],
"density": 1,
"position": [
631,
772
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
922.18,
-69.9
]
},
{
"color": [
224,
212,
248
],
"density": 1,
"position": [
783,
632
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-337.33,
926.02
]
},
{
"color": [
251,
169,
85
],
"density": 1,
"position": [
803,
890
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-935.52,
-277.57
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 756,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3635719
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (729, 820), initial velocity (-956.23, -275.59) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (690, 954), initial velocity (-98.49, 840.56) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (845, 952), initial velocity (232.5, 826.18) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (604, 750), initial velocity (626.55, 561.94) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (944, 682), initial velocity (542.9, 798.98) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (945, 786), initial velocity (305.57, 907.24) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[714.61, 839.34], [690.39, 1027.82], [813.05, 979.42], [626.04, 782.74], [998.3, 761.91], [970.94, 872.1]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[825.62, 931.09], [774.45, 1017.98], [730.43, 956.18], [585.39, 779.28], [1005.85, 795.07], [880.22, 841.53]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[831.92, 918.08], [846.87, 996.27], [647.61, 932.88], [544.64, 775.8], [925.75, 740.64], [789.27, 810.89]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[740.17, 807.07], [856.71, 912.21], [564.99, 909.64], [503.99, 772.34], [845.84, 686.34], [698.55, 780.32]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[648.42, 696.06], [866.55, 828.15], [538.73, 830.04], [463.32, 768.87], [765.93, 632.05], [607.83, 749.76]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[567.29, 595.66], [876.39, 744.1], [515.49, 753.84], [422.66, 765.4], [686.03, 577.75], [517.1, 712.77]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[678.3, 687.41], [886.23, 660.04], [492.24, 760.6], [422.66, 738.06], [628.88, 546.22], [522.18, 688.62]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[789.31, 779.15], [896.07, 575.98], [503.66, 816.65], [463.32, 741.53], [683.18, 626.12], [604.8, 658.06]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[900.32, 870.9], [846.91, 550.92], [594.38, 793.41], [503.98, 744.99], [737.48, 706.03], [687.42, 627.49]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[892.34, 843.66], [762.85, 560.76], [685.11, 770.17], [544.64, 748.46], [791.77, 785.94], [770.03, 596.93]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[807.59, 732.65], [678.8, 570.6], [772.33, 746.92], [585.28, 751.93], [846.07, 865.85], [852.65, 566.36]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[959.13, 621.64], [594.74, 580.44], [741.42, 723.68], [625.93, 755.39], [900.37, 945.75], [883.46, 587.6]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[865.89, 755.3], [562.08, 641.44], [710.59, 700.5], [666.48, 758.85], [822.21, 893.14], [852.97, 670.01]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[821.57, 868.33], [552.24, 725.5], [679.68, 677.26], [707.13, 762.32], [708.95, 858.09], [822.4, 752.63]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[860.51, 933.3], [542.4, 809.56], [648.77, 654.01], [747.78, 765.79], [554.07, 847.08], [791.84, 835.25]]) <= 50.0"
] | ballsim | G_3_718 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
65,
199,
222
],
"density": 1,
"position": [
729,
820
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-956.23,
-275.59
]
},
{
"color": [
233,
135,
139
],
"density": 1,
"position": [
690,
954
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-98.49,
840.56
]
},
{
"color": [
234,
204,
96
],
"density": 1,
"position": [
845,
952
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
232.5,
826.18
]
},
{
"color": [
153,
171,
80
],
"density": 1,
"position": [
604,
750
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
626.55,
561.94
]
},
{
"color": [
155,
216,
161
],
"density": 1,
"position": [
944,
682
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
542.9,
798.98
]
},
{
"color": [
56,
100,
213
],
"density": 1,
"position": [
945,
786
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
305.57,
907.24
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 718,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6173837
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (830, 856), initial velocity (-986.92, -118.31) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (704, 908), initial velocity (474.91, -647.92) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (756, 769), initial velocity (65.0, 892.65) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (723, 604), initial velocity (-881.02, 398.71) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (910, 872), initial velocity (-486.65, -782.89) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (785, 714), initial velocity (816.11, -508.99) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[780.95, 860.01], [690.75, 878.27], [723.95, 791.51], [634.89, 643.87], [861.34, 793.7], [866.62, 663.1]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[817.35, 878.08], [567.86, 909.14], [637.01, 718.95], [546.78, 683.74], [812.69, 715.41], [948.23, 612.19]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[853.85, 896.19], [605.21, 779.55], [549.86, 646.21], [458.46, 723.7], [763.91, 636.92], [908.83, 682.39]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[890.25, 914.26], [642.47, 650.27], [578.68, 689.41], [434.31, 736.43], [715.25, 558.62], [857.92, 764.0]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[915.35, 921.02], [679.72, 521.0], [651.24, 776.35], [522.41, 696.56], [672.62, 486.36], [807.02, 845.62]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[897.28, 884.62], [654.87, 582.84], [723.8, 863.29], [610.52, 656.69], [780.39, 435.06], [756.12, 927.24]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[879.21, 848.22], [629.56, 560.84], [796.36, 950.23], [698.63, 659.46], [723.85, 547.53], [705.22, 1008.85]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[861.15, 811.82], [665.76, 548.09], [817.82, 986.07], [786.74, 688.37], [667.3, 660.01], [742.8, 1001.98]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[843.08, 775.41], [740.03, 573.41], [842.72, 848.94], [874.85, 717.28], [610.76, 772.48], [768.49, 983.37]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[825.01, 739.01], [814.3, 598.73], [911.81, 686.29], [962.96, 746.18], [554.22, 884.96], [772.08, 977.51]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[806.95, 702.61], [848.62, 624.05], [918.58, 625.92], [1051.07, 775.09], [650.38, 844.73], [775.66, 971.65]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[788.88, 666.21], [685.97, 649.36], [992.84, 695.01], [1043.67, 708.49], [762.86, 788.18], [779.25, 965.79]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[770.86, 629.9], [523.73, 674.62], [1066.92, 763.93], [968.27, 667.17], [875.05, 731.78], [782.83, 959.94]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[752.79, 593.5], [452.42, 791.26], [1021.97, 713.79], [880.16, 638.27], [987.53, 675.24], [786.42, 954.08]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[734.73, 557.1], [606.24, 774.77], [877.63, 769.82], [792.05, 609.36], [1009.57, 679.03], [790.01, 948.23]]) <= 50.0"
] | ballsim | G_3_396 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
231,
183,
115
],
"density": 1,
"position": [
830,
856
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-986.92,
-118.31
]
},
{
"color": [
130,
204,
134
],
"density": 1,
"position": [
704,
908
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
474.91,
-647.92
]
},
{
"color": [
142,
130,
247
],
"density": 1,
"position": [
756,
769
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
65,
892.65
]
},
{
"color": [
243,
229,
63
],
"density": 1,
"position": [
723,
604
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-881.02,
398.71
]
},
{
"color": [
52,
141,
123
],
"density": 1,
"position": [
910,
872
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-486.65,
-782.89
]
},
{
"color": [
111,
231,
142
],
"density": 1,
"position": [
785,
714
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
816.11,
-508.99
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 396,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7933787
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (599, 794), initial velocity (703.29, -514.97) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (680, 746), initial velocity (-965.76, 17.18) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (814, 970), initial velocity (671.39, 640.61) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (749, 937), initial velocity (601.56, -680.1) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (559, 703), initial velocity (363.77, 717.31) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[581.84, 817.18], [666.17, 687.6], [881.14, 1034.06], [809.16, 868.98], [592.46, 765.75]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[528.05, 918.21], [698.18, 595.9], [822.33, 1006.62], [869.31, 800.96], [612.45, 787.04]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[507.72, 916.1], [730.26, 503.97], [740.47, 962.44], [929.62, 732.78], [632.5, 808.38]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[523.59, 802.74], [783.04, 476.18], [658.8, 918.37], [989.78, 664.76], [652.5, 829.67]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[539.46, 689.39], [862.85, 531.55], [577.14, 874.31], [972.86, 652.74], [672.49, 850.96]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[555.33, 576.03], [942.66, 586.92], [495.47, 830.24], [889.59, 688.95], [692.49, 872.25]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[596.74, 541.27], [1022.47, 642.29], [511.83, 786.17], [806.31, 725.16], [712.48, 893.53]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[676.21, 623.64], [1052.7, 733.69], [593.49, 742.1], [723.03, 761.36], [732.48, 914.82]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[755.67, 706.02], [1036.17, 794.32], [675.16, 698.04], [639.76, 797.57], [752.47, 936.11]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[835.14, 788.39], [939.05, 796.42], [756.82, 653.97], [556.48, 833.77], [772.47, 957.4]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[871.7, 901.93], [881.2, 769.99], [838.49, 609.9], [473.2, 869.98], [792.46, 978.69]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[870.33, 1043.03], [858.06, 718.36], [920.15, 565.83], [543.33, 906.19], [812.46, 999.98]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[786.25, 1044.28], [834.97, 666.85], [919.98, 581.2], [626.4, 942.3], [841.82, 992.21]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[674.09, 1022.64], [811.83, 615.21], [852.84, 645.26], [709.67, 978.51], [892.11, 920.19]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[567.9, 982.66], [788.68, 563.58], [785.7, 709.32], [792.95, 1014.71], [942.4, 848.17]]) <= 50.0"
] | ballsim | G_3_829 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
177,
226,
148
],
"density": 1,
"position": [
599,
794
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
703.29,
-514.97
]
},
{
"color": [
246,
155,
124
],
"density": 1,
"position": [
680,
746
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-965.76,
17.18
]
},
{
"color": [
190,
84,
118
],
"density": 1,
"position": [
814,
970
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
671.39,
640.61
]
},
{
"color": [
190,
138,
114
],
"density": 1,
"position": [
749,
937
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
601.56,
-680.1
]
},
{
"color": [
79,
137,
193
],
"density": 1,
"position": [
559,
703
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
363.77,
717.31
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 829,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2256553
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (867, 925), initial velocity (-635.31, -644.05) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (570, 777), initial velocity (-868.67, 155.76) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (724, 739), initial velocity (679.05, -538.78) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (906, 772), initial velocity (322.42, -917.97) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (656, 641), initial velocity (98.03, 880.43) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (906, 605), initial velocity (857.31, 282.54) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[803.47, 860.6], [483.13, 792.58], [791.9, 685.12], [938.25, 680.2], [665.81, 729.04], [991.74, 633.25]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[739.95, 796.19], [529.35, 808.15], [859.79, 631.24], [970.5, 588.41], [675.63, 817.07], [1029.32, 696.48]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[676.27, 731.63], [616.43, 823.77], [893.61, 611.47], [936.07, 533.75], [685.47, 905.33], [1029.72, 786.97]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[612.74, 667.22], [703.3, 839.34], [860.38, 658.71], [832.28, 507.92], [695.28, 993.37], [1007.13, 860.54]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[549.21, 602.82], [790.16, 854.92], [827.15, 705.95], [713.04, 493.3], [730.98, 1001.76], [921.16, 888.08]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[485.69, 538.41], [877.03, 870.5], [793.93, 753.19], [607.36, 520.44], [790.67, 936.31], [835.2, 915.61]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[471.26, 625.12], [963.89, 886.07], [760.7, 800.43], [519.47, 602.35], [720.52, 965.17], [879.08, 848.81]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[484.54, 741.4], [977.52, 848.43], [727.47, 847.67], [474.84, 657.43], [640.66, 1001.08], [932.67, 774.96]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[517.94, 877.8], [935.87, 770.63], [694.24, 894.92], [515.79, 692.39], [588.13, 952.87], [986.26, 701.11]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[568.84, 960.36], [894.22, 692.82], [661.01, 942.16], [556.73, 727.35], [544.62, 876.87], [1008.38, 650.12]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[676.04, 869.66], [852.57, 615.01], [627.79, 989.4], [597.67, 762.31], [501.11, 800.87], [921.59, 678.27]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[783.24, 778.96], [810.92, 537.2], [615.25, 973.0], [638.61, 797.28], [468.01, 724.87], [834.8, 706.42]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[848.93, 815.38], [769.38, 459.59], [616.15, 915.38], [679.45, 832.15], [511.41, 649.06], [782.9, 627.74]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[898.05, 903.36], [782.31, 479.64], [617.05, 857.62], [720.4, 867.11], [554.91, 573.06], [733.54, 540.62]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[924.21, 974.67], [864.68, 433.77], [617.95, 799.85], [761.34, 902.07], [598.55, 497.46], [638.76, 593.33]]) <= 50.0"
] | ballsim | G_3_403 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
230,
112,
180
],
"density": 1,
"position": [
867,
925
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-635.31,
-644.05
]
},
{
"color": [
236,
180,
165
],
"density": 1,
"position": [
570,
777
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-868.67,
155.76
]
},
{
"color": [
209,
136,
156
],
"density": 1,
"position": [
724,
739
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
679.05,
-538.78
]
},
{
"color": [
110,
234,
209
],
"density": 1,
"position": [
906,
772
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
322.42,
-917.97
]
},
{
"color": [
158,
230,
248
],
"density": 1,
"position": [
656,
641
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
98.03,
880.43
]
},
{
"color": [
118,
82,
102
],
"density": 1,
"position": [
906,
605
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
857.31,
282.54
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 403,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3740715
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (1007, 786), initial velocity (-196.02, -918.44) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (802, 569), initial velocity (486.3, 788.67) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (957, 584), initial velocity (-36.94, -867.96) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (860, 656), initial velocity (-121.2, 854.7) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (819, 892), initial velocity (-370.13, -829.52) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (651, 705), initial velocity (-630.44, -770.76) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[987.4, 694.15], [850.63, 647.86], [908.05, 530.09], [847.89, 741.47], [781.99, 809.04], [587.96, 627.92]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[967.79, 602.31], [899.27, 726.71], [826.65, 499.75], [835.78, 826.95], [744.98, 726.08], [524.93, 550.85]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[922.04, 529.19], [948.02, 805.77], [745.05, 469.34], [823.64, 912.63], [707.87, 642.92], [503.52, 602.21]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[840.74, 482.17], [996.65, 884.63], [687.54, 512.52], [811.53, 998.11], [670.86, 559.96], [497.82, 701.62]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[767.3, 459.33], [920.39, 872.76], [663.29, 520.2], [810.65, 1049.02], [633.85, 627.66], [492.12, 801.03]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[729.16, 545.15], [830.35, 850.88], [669.15, 609.97], [851.08, 972.75], [596.84, 706.93], [486.41, 900.45]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[691.03, 630.97], [740.31, 829.01], [675.01, 699.74], [891.5, 896.48], [559.83, 786.2], [503.27, 930.45]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[652.89, 716.79], [650.27, 807.13], [680.87, 789.51], [931.93, 820.21], [540.79, 810.13], [535.73, 912.46]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[614.76, 802.6], [560.23, 785.26], [686.73, 879.28], [972.36, 743.94], [555.49, 730.25], [552.19, 943.7]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[576.63, 825.93], [470.19, 763.38], [692.59, 969.05], [1012.79, 667.67], [570.19, 650.37], [607.97, 935.1]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[542.55, 766.01], [524.98, 727.44], [714.41, 1009.68], [966.77, 654.22], [584.89, 570.48], [677.15, 1002.52]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[538.79, 758.59], [546.27, 586.5], [809.89, 962.44], [881.73, 669.12], [603.29, 502.0], [736.8, 918.89]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[535.04, 751.19], [605.7, 563.45], [906.04, 915.82], [796.91, 683.97], [661.98, 557.84], [795.12, 834.8]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[531.28, 743.77], [705.77, 664.96], [1002.43, 869.09], [711.88, 698.87], [720.82, 613.82], [853.59, 750.5]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[527.52, 736.35], [651.9, 651.67], [1034.68, 775.76], [681.9, 809.1], [803.61, 628.33], [912.07, 666.2]]) <= 50.0"
] | ballsim | G_3_122 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
243,
73,
57
],
"density": 1,
"position": [
1007,
786
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-196.02,
-918.44
]
},
{
"color": [
73,
176,
213
],
"density": 1,
"position": [
802,
569
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
486.3,
788.67
]
},
{
"color": [
198,
76,
111
],
"density": 1,
"position": [
957,
584
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-36.94,
-867.96
]
},
{
"color": [
204,
163,
113
],
"density": 1,
"position": [
860,
656
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-121.2,
854.7
]
},
{
"color": [
245,
114,
155
],
"density": 1,
"position": [
819,
892
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-370.13,
-829.52
]
},
{
"color": [
72,
238,
205
],
"density": 1,
"position": [
651,
705
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-630.44,
-770.76
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 122,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 9477934
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (793, 743), initial velocity (-851.62, 326.36) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (846, 767), initial velocity (-219.73, -805.62) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (734, 806), initial velocity (-978.01, 194.15) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (744, 685), initial velocity (-942.95, 292.01) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (727, 740), initial velocity (968.82, -58.24) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (663, 684), initial velocity (805.93, 16.89) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[773.89, 813.78], [824.03, 686.43], [636.2, 825.41], [713.76, 610.55], [751.61, 761.54], [619.7, 685.68]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[768.64, 892.56], [802.05, 605.87], [626.62, 844.82], [672.16, 536.11], [779.66, 821.07], [619.6, 687.37]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[732.32, 917.74], [740.41, 593.7], [724.67, 864.28], [589.59, 591.45], [807.78, 880.74], [714.13, 689.06]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[661.47, 882.91], [668.73, 599.22], [795.29, 867.98], [692.04, 670.86], [835.83, 877.46], [808.41, 690.74]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[590.62, 848.07], [615.21, 573.27], [804.22, 836.08], [766.53, 798.72], [863.88, 731.73], [902.7, 692.43]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[645.25, 813.23], [603.32, 547.31], [852.16, 871.75], [741.05, 753.43], [856.44, 647.5], [996.99, 694.11]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[716.1, 778.39], [656.84, 521.36], [827.21, 781.18], [701.45, 683.68], [744.28, 744.67], [1051.21, 765.23]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[661.16, 816.17], [679.86, 548.21], [827.55, 734.41], [661.85, 613.93], [833.09, 608.92], [1019.21, 706.46]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[602.39, 856.16], [684.13, 607.51], [842.24, 712.51], [622.25, 544.18], [716.27, 738.68], [930.1, 699.18]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[621.39, 916.68], [688.4, 666.81], [822.85, 690.61], [582.66, 474.43], [585.68, 851.24], [904.0, 700.86]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[638.0, 959.38], [692.67, 726.11], [764.89, 668.71], [590.6, 487.04], [689.93, 853.34], [955.02, 702.55]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[597.87, 850.62], [696.95, 785.41], [706.93, 646.81], [631.2, 556.21], [820.52, 855.43], [1006.05, 704.23]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[668.54, 742.12], [701.21, 844.57], [662.95, 648.94], [644.03, 577.28], [934.76, 829.76], [1042.8, 730.45]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[739.39, 633.36], [705.48, 903.87], [647.69, 701.04], [599.21, 498.47], [998.24, 715.6], [1066.77, 768.56]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[752.41, 624.76], [709.75, 963.17], [632.43, 753.14], [583.28, 469.73], [954.7, 786.8], [1042.69, 723.52]]) <= 50.0"
] | ballsim | G_3_626 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
58,
99,
122
],
"density": 1,
"position": [
793,
743
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-851.62,
326.36
]
},
{
"color": [
232,
121,
179
],
"density": 1,
"position": [
846,
767
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-219.73,
-805.62
]
},
{
"color": [
220,
235,
245
],
"density": 1,
"position": [
734,
806
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-978.01,
194.15
]
},
{
"color": [
104,
69,
106
],
"density": 1,
"position": [
744,
685
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-942.95,
292.01
]
},
{
"color": [
251,
230,
186
],
"density": 1,
"position": [
727,
740
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
968.82,
-58.24
]
},
{
"color": [
123,
126,
165
],
"density": 1,
"position": [
663,
684
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
805.93,
16.89
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 626,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1603914
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (765, 741), initial velocity (216.89, -956.69) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (731, 823), initial velocity (-477.29, 655.91) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (535, 778), initial velocity (-587.93, -676.54) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (837, 694), initial velocity (675.44, -591.73) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (734, 991), initial velocity (555.81, -682.31) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[786.68, 645.32], [683.27, 888.6], [476.21, 710.35], [904.55, 634.82], [789.59, 922.76]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[808.36, 549.64], [635.54, 954.2], [515.84, 642.7], [972.11, 575.64], [845.18, 854.53]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[830.09, 453.73], [601.64, 977.05], [574.77, 574.88], [902.88, 615.81], [900.91, 786.12]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[887.31, 467.42], [601.59, 895.92], [633.56, 507.23], [825.73, 661.75], [956.5, 717.88]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[942.2, 545.8], [601.54, 814.79], [710.09, 494.16], [748.59, 707.7], [1012.09, 649.64]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[970.74, 618.36], [601.49, 733.66], [797.42, 514.34], [671.44, 753.65], [962.78, 673.84]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[942.18, 592.04], [601.44, 652.54], [884.75, 534.5], [594.29, 799.59], [924.4, 781.26]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[913.61, 565.72], [601.39, 571.41], [960.29, 563.23], [517.14, 845.54], [886.03, 888.69]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[885.05, 539.4], [606.65, 506.63], [952.5, 652.51], [493.26, 891.49], [847.65, 996.11]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[856.48, 513.08], [654.29, 572.3], [944.72, 741.8], [570.41, 937.44], [830.01, 1039.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[827.92, 486.75], [701.92, 637.97], [936.93, 831.08], [647.56, 983.38], [862.09, 930.22]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[799.35, 460.42], [749.55, 703.65], [929.14, 920.36], [724.71, 1029.33], [894.17, 820.74]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[777.12, 453.42], [797.06, 769.16], [915.01, 1004.8], [807.19, 1058.16], [926.17, 711.54]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[769.48, 491.51], [844.7, 834.83], [871.3, 1000.99], [870.22, 1066.92], [958.25, 602.07]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[761.83, 529.59], [892.33, 900.5], [888.17, 936.61], [925.37, 974.05], [895.61, 561.41]]) <= 50.0"
] | ballsim | G_3_628 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
124,
94,
232
],
"density": 1,
"position": [
765,
741
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
216.89,
-956.69
]
},
{
"color": [
166,
62,
108
],
"density": 1,
"position": [
731,
823
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-477.29,
655.91
]
},
{
"color": [
53,
123,
190
],
"density": 1,
"position": [
535,
778
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-587.93,
-676.54
]
},
{
"color": [
226,
138,
131
],
"density": 1,
"position": [
837,
694
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
675.44,
-591.73
]
},
{
"color": [
123,
132,
56
],
"density": 1,
"position": [
734,
991
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
555.81,
-682.31
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 628,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2095023
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (709, 731), initial velocity (763.08, 362.54) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (767, 716), initial velocity (-194.55, 903.48) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (558, 654), initial velocity (-911.56, -120.61) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (685, 807), initial velocity (824.05, 179.98) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (734, 971), initial velocity (523.13, 802.83) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[731.94, 767.25], [829.72, 806.36], [503.07, 678.18], [767.4, 824.99], [786.32, 1051.27]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[732.85, 803.51], [926.38, 896.71], [515.14, 769.34], [849.79, 842.99], [723.44, 1016.37]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[733.75, 839.86], [911.57, 907.71], [527.23, 860.73], [888.53, 785.03], [642.97, 963.92]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[734.66, 876.11], [883.15, 940.92], [571.61, 868.99], [878.1, 642.23], [602.88, 949.27]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[735.56, 912.37], [819.01, 938.43], [569.9, 788.2], [854.42, 512.69], [693.53, 1011.94]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[727.33, 948.62], [768.94, 935.94], [568.19, 707.41], [711.62, 502.27], [780.35, 1070.8]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[677.01, 984.87], [783.66, 933.45], [566.48, 626.63], [660.49, 583.51], [717.68, 980.17]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[666.14, 981.68], [798.38, 930.96], [599.17, 580.24], [670.92, 726.31], [655.01, 889.52]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[702.4, 931.36], [813.1, 928.47], [679.96, 581.95], [681.34, 869.1], [592.33, 798.87]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[787.47, 929.86], [827.82, 925.98], [760.74, 583.66], [616.6, 936.74], [529.66, 708.22]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[828.89, 945.42], [931.49, 919.0], [841.53, 585.37], [630.45, 873.28], [534.77, 685.34]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[862.17, 960.97], [933.98, 798.3], [922.32, 587.07], [671.84, 782.28], [625.42, 748.01]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[878.51, 959.62], [936.47, 677.89], [943.77, 647.92], [745.85, 691.52], [683.1, 810.53]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[862.95, 926.35], [834.31, 738.44], [965.98, 658.44], [836.5, 600.53], [724.49, 873.2]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[847.4, 893.07], [730.77, 801.38], [927.5, 728.99], [864.52, 572.16], [765.87, 935.87]]) <= 50.0"
] | ballsim | G_3_755 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
254,
120,
174
],
"density": 1,
"position": [
709,
731
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
763.08,
362.54
]
},
{
"color": [
238,
213,
192
],
"density": 1,
"position": [
767,
716
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-194.55,
903.48
]
},
{
"color": [
97,
79,
56
],
"density": 1,
"position": [
558,
654
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-911.56,
-120.61
]
},
{
"color": [
102,
206,
212
],
"density": 1,
"position": [
685,
807
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
824.05,
179.98
]
},
{
"color": [
117,
98,
76
],
"density": 1,
"position": [
734,
971
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
523.13,
802.83
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 755,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3502467
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (771, 600), initial velocity (215.39, 825.95) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (862, 740), initial velocity (556.07, -803.46) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (782, 864), initial velocity (-325.56, -845.78) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (604, 691), initial velocity (522.55, 791.49) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (683, 608), initial velocity (815.99, 552.88) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[794.35, 683.91], [917.62, 659.65], [749.46, 779.43], [656.25, 770.15], [762.78, 661.98]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[842.2, 785.62], [973.23, 579.31], [716.91, 694.86], [708.49, 849.3], [818.05, 698.14]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[890.17, 887.58], [889.53, 600.07], [684.29, 610.08], [760.87, 928.65], [873.46, 734.38]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[929.0, 982.74], [795.92, 628.12], [651.74, 525.51], [813.11, 1007.8], [928.74, 770.54]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[817.48, 968.65], [702.32, 656.17], [650.43, 537.09], [855.19, 1079.56], [984.01, 806.7]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[705.96, 954.56], [608.72, 684.22], [673.8, 624.64], [764.61, 1051.72], [1036.98, 841.18]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[594.43, 940.48], [515.11, 712.27], [697.16, 712.19], [675.82, 1018.42], [985.52, 799.78]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[482.91, 926.39], [491.74, 740.33], [720.52, 799.74], [587.03, 985.09], [934.05, 758.37]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[554.24, 912.3], [585.34, 768.38], [743.89, 887.29], [498.23, 951.77], [882.59, 716.96]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[665.77, 898.22], [678.94, 796.43], [767.25, 974.83], [523.82, 918.44], [831.12, 675.56]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[777.29, 884.13], [772.54, 824.48], [791.07, 1060.99], [612.61, 885.12], [779.66, 634.15]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[889.45, 871.14], [864.99, 850.53], [861.46, 1003.9], [701.41, 851.79], [728.19, 592.75]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[1010.41, 873.89], [940.61, 847.75], [931.67, 946.94], [789.98, 818.55], [676.86, 551.44]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[1030.21, 763.76], [917.72, 844.97], [1002.05, 889.83], [878.77, 785.22], [625.39, 510.04]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[1019.39, 687.51], [843.62, 842.19], [1036.36, 806.54], [967.57, 751.9], [595.43, 534.85]]) <= 50.0"
] | ballsim | G_3_18 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
80,
60,
249
],
"density": 1,
"position": [
771,
600
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
215.39,
825.95
]
},
{
"color": [
87,
232,
164
],
"density": 1,
"position": [
862,
740
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
556.07,
-803.46
]
},
{
"color": [
96,
184,
227
],
"density": 1,
"position": [
782,
864
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-325.56,
-845.78
]
},
{
"color": [
121,
186,
250
],
"density": 1,
"position": [
604,
691
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
522.55,
791.49
]
},
{
"color": [
57,
213,
204
],
"density": 1,
"position": [
683,
608
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
815.99,
552.88
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 18,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8357688
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (724, 635), initial velocity (4.96, 922.95) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (783, 653), initial velocity (566.99, -583.55) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (792, 719), initial velocity (-874.07, 381.24) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (880, 832), initial velocity (49.88, 896.78) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (799, 784), initial velocity (-805.09, -560.38) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[730.47, 708.89], [830.48, 610.61], [669.49, 748.96], [836.74, 838.11], [742.65, 769.81]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[747.71, 749.68], [808.28, 688.88], [632.61, 813.68], [761.57, 788.94], [701.2, 781.41]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[731.3, 686.95], [786.04, 767.34], [783.0, 900.36], [723.75, 831.38], [659.64, 793.03]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[713.26, 619.23], [716.26, 798.03], [838.95, 843.52], [696.01, 853.25], [618.18, 804.63]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[695.22, 551.51], [584.18, 763.07], [961.94, 828.82], [668.28, 875.13], [576.73, 816.23]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[650.96, 529.18], [719.61, 728.11], [1032.91, 724.01], [640.55, 897.0], [609.75, 827.82]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[583.29, 547.42], [855.03, 693.15], [991.17, 814.45], [612.81, 918.88], [651.2, 839.42]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[641.77, 565.66], [960.89, 709.4], [866.01, 807.28], [585.08, 940.75], [692.66, 851.02]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[709.45, 583.9], [962.79, 782.59], [739.77, 798.22], [600.05, 962.63], [734.11, 862.61]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[777.12, 602.13], [864.82, 682.78], [613.52, 789.17], [627.79, 984.5], [775.57, 874.21]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[752.86, 579.2], [854.05, 670.18], [657.74, 780.11], [648.65, 994.49], [817.02, 885.81]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[625.44, 581.26], [910.15, 724.45], [783.99, 771.05], [643.55, 959.53], [854.08, 889.77]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[659.04, 583.3], [966.11, 778.59], [909.92, 762.01], [638.46, 924.65], [864.74, 848.18]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[786.46, 585.35], [1000.47, 795.18], [1011.61, 710.42], [633.36, 889.69], [875.44, 806.48]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[859.05, 682.36], [1011.06, 770.62], [921.26, 802.79], [628.25, 854.73], [886.13, 764.78]]) <= 50.0"
] | ballsim | G_3_636 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
172,
210,
161
],
"density": 1,
"position": [
724,
635
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
4.96,
922.95
]
},
{
"color": [
145,
130,
255
],
"density": 1,
"position": [
783,
653
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
566.99,
-583.55
]
},
{
"color": [
127,
120,
143
],
"density": 1,
"position": [
792,
719
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-874.07,
381.24
]
},
{
"color": [
244,
203,
187
],
"density": 1,
"position": [
880,
832
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
49.88,
896.78
]
},
{
"color": [
212,
246,
155
],
"density": 1,
"position": [
799,
784
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-805.09,
-560.38
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 636,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7780023
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (914, 649), initial velocity (523.61, -747.06) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (611, 682), initial velocity (339.13, -938.65) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (771, 773), initial velocity (-697.21, -468.51) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (648, 799), initial velocity (-585.76, -632.69) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (802, 849), initial velocity (-828.36, -468.35) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (684, 763), initial velocity (594.08, 657.35) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[929.28, 611.38], [644.91, 588.13], [701.27, 726.15], [589.43, 735.74], [758.96, 825.15], [663.81, 782.79]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[854.57, 663.75], [678.82, 494.26], [631.55, 679.3], [530.86, 672.49], [779.73, 838.13], [515.98, 728.89]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[779.68, 716.25], [744.86, 432.2], [561.65, 632.33], [543.88, 680.81], [800.56, 851.16], [477.5, 784.55]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[704.97, 768.62], [753.27, 483.73], [566.85, 660.42], [607.14, 739.37], [821.34, 864.14], [584.71, 879.07]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[686.13, 861.57], [719.36, 577.6], [613.7, 730.14], [639.87, 775.77], [842.11, 877.13], [732.53, 932.98]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[732.06, 1001.56], [685.45, 671.48], [627.23, 742.13], [655.45, 818.01], [862.89, 890.12], [868.1, 974.62]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[729.35, 1061.1], [672.0, 744.87], [593.52, 758.38], [679.62, 875.14], [904.13, 867.69], [773.29, 897.65]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[683.42, 921.11], [693.54, 783.28], [505.94, 828.5], [645.58, 932.27], [958.89, 821.84], [757.97, 867.53]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[637.5, 781.12], [715.07, 821.68], [550.95, 766.04], [642.35, 913.61], [1013.65, 775.99], [824.98, 837.4]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[651.08, 641.13], [736.6, 860.08], [561.57, 678.46], [699.48, 834.58], [1068.39, 730.14], [892.0, 807.27]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[721.19, 501.14], [758.14, 898.49], [561.44, 636.69], [756.61, 755.55], [1031.96, 775.48], [959.02, 777.14]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[716.13, 436.33], [779.67, 936.89], [649.02, 682.61], [813.74, 676.52], [986.11, 830.24], [1026.03, 747.02]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[646.07, 575.85], [801.15, 975.2], [736.37, 728.42], [870.72, 597.69], [940.37, 884.87], [1071.95, 737.88]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[575.95, 715.84], [822.68, 1013.6], [823.94, 774.34], [868.45, 578.07], [894.52, 939.63], [1041.82, 804.89]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[508.17, 853.5], [802.98, 1010.77], [911.52, 820.27], [789.42, 635.2], [848.67, 994.39], [983.08, 843.29]]) <= 50.0"
] | ballsim | G_3_1068 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
199,
229,
207
],
"density": 1,
"position": [
914,
649
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
523.61,
-747.06
]
},
{
"color": [
236,
216,
129
],
"density": 1,
"position": [
611,
682
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
339.13,
-938.65
]
},
{
"color": [
196,
62,
164
],
"density": 1,
"position": [
771,
773
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-697.21,
-468.51
]
},
{
"color": [
180,
131,
103
],
"density": 1,
"position": [
648,
799
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-585.76,
-632.69
]
},
{
"color": [
107,
158,
172
],
"density": 1,
"position": [
802,
849
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-828.36,
-468.35
]
},
{
"color": [
65,
230,
98
],
"density": 1,
"position": [
684,
763
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
594.08,
657.35
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1068,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 563393
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (730, 790), initial velocity (960.84, 179.07) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (562, 948), initial velocity (130.01, -894.44) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (618, 846), initial velocity (-466.34, 788.92) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (823, 698), initial velocity (-803.13, 192.82) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (824, 795), initial velocity (481.83, 741.31) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (874, 808), initial velocity (161.34, 861.57) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[826.09, 807.9], [513.55, 894.04], [632.82, 889.4], [742.68, 717.29], [869.49, 866.42], [892.41, 896.43]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[922.19, 825.79], [494.14, 823.31], [703.79, 900.39], [662.36, 736.57], [906.8, 929.68], [917.7, 991.74]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[1018.52, 843.73], [527.89, 721.69], [774.94, 911.4], [581.83, 755.91], [944.19, 993.1], [893.86, 1052.56]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1003.49, 797.48], [561.55, 620.32], [845.92, 922.39], [501.51, 775.2], [901.42, 1010.12], [798.69, 1026.78]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[939.94, 723.21], [595.22, 518.95], [916.89, 933.38], [421.2, 794.48], [827.98, 1009.44], [703.53, 1001.02]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[876.39, 648.94], [628.89, 467.56], [987.86, 944.36], [477.3, 735.01], [754.55, 1008.75], [608.36, 975.27]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[812.84, 574.68], [662.55, 568.92], [950.24, 892.64], [534.16, 675.1], [681.11, 1008.07], [513.19, 949.51]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[749.29, 500.41], [696.22, 670.29], [905.24, 836.66], [591.02, 615.19], [607.67, 1007.38], [538.44, 854.22]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[685.74, 464.34], [729.89, 771.66], [860.25, 780.67], [647.88, 555.27], [547.71, 998.92], [563.74, 758.91]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[617.65, 534.07], [763.56, 873.03], [815.25, 724.69], [708.23, 498.85], [583.84, 934.98], [589.03, 663.6]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[546.31, 600.55], [797.22, 974.39], [770.26, 668.71], [771.07, 444.92], [619.98, 871.04], [614.32, 568.28]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[474.98, 667.03], [830.89, 1039.1], [725.26, 612.73], [833.91, 494.15], [656.11, 807.1], [639.62, 472.98]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[405.03, 734.03], [864.47, 937.98], [680.38, 556.89], [896.6, 547.95], [692.15, 743.32], [664.85, 510.63]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[469.41, 779.23], [898.14, 836.62], [635.38, 636.02], [959.44, 601.88], [728.28, 679.38], [690.14, 458.29]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[562.65, 807.77], [931.81, 735.25], [590.39, 724.62], [1022.28, 655.81], [764.42, 615.44], [715.43, 492.93]]) <= 50.0"
] | ballsim | G_3_762 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
212,
53,
104
],
"density": 1,
"position": [
730,
790
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
960.84,
179.07
]
},
{
"color": [
235,
205,
197
],
"density": 1,
"position": [
562,
948
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
130.01,
-894.44
]
},
{
"color": [
216,
131,
133
],
"density": 1,
"position": [
618,
846
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-466.34,
788.92
]
},
{
"color": [
176,
157,
242
],
"density": 1,
"position": [
823,
698
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-803.13,
192.82
]
},
{
"color": [
50,
238,
138
],
"density": 1,
"position": [
824,
795
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
481.83,
741.31
]
},
{
"color": [
145,
188,
159
],
"density": 1,
"position": [
874,
808
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
161.34,
861.57
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 762,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6404776
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (693, 712), initial velocity (-731.81, -530.41) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (748, 656), initial velocity (-509.77, -784.74) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (1014, 772), initial velocity (-183.86, 858.73) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (619, 592), initial velocity (657.54, -640.71) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (601, 676), initial velocity (-931.3, 251.89) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (856, 789), initial velocity (953.81, 19.61) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[619.83, 658.95], [697.02, 577.53], [995.62, 857.86], [684.75, 527.94], [507.86, 701.2], [951.39, 790.95]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[546.66, 605.9], [646.05, 499.06], [977.23, 943.73], [750.49, 463.87], [414.73, 726.39], [1046.77, 792.91]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[526.72, 583.56], [594.94, 468.15], [935.75, 1016.51], [816.41, 490.87], [433.25, 812.95], [1047.45, 740.06]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[609.24, 620.42], [543.97, 546.63], [870.57, 1039.51], [882.15, 554.94], [501.63, 881.02], [998.07, 658.44]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[691.76, 657.29], [492.99, 625.09], [805.38, 980.65], [894.01, 587.89], [570.02, 949.08], [990.17, 600.77]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[774.28, 694.15], [442.01, 703.56], [740.2, 921.79], [790.89, 554.47], [638.4, 1017.15], [987.17, 642.48]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[856.79, 731.02], [429.19, 760.0], [675.01, 862.93], [687.76, 521.05], [706.79, 1029.64], [941.17, 709.0]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[850.74, 819.01], [522.63, 755.1], [609.83, 804.07], [584.64, 487.61], [775.17, 961.57], [939.46, 749.97]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[836.71, 854.91], [535.66, 691.76], [618.24, 798.67], [607.95, 527.18], [794.45, 978.56], [966.29, 774.46]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[888.3, 777.15], [493.89, 588.61], [676.81, 829.7], [688.46, 599.76], [748.11, 1005.64], [993.13, 798.94]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[939.89, 699.39], [600.0, 570.83], [735.38, 860.73], [768.98, 672.34], [701.77, 875.0], [1019.96, 823.43]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[991.47, 621.63], [710.2, 555.43], [793.95, 891.76], [849.5, 744.92], [655.43, 744.37], [1046.81, 847.92]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[957.67, 593.3], [820.13, 540.06], [852.38, 922.71], [929.82, 817.33], [609.21, 614.05], [1014.73, 838.38]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[864.53, 599.11], [930.33, 524.66], [910.94, 953.74], [1001.85, 898.41], [562.87, 483.41], [986.65, 820.86]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[771.39, 604.92], [928.71, 573.82], [969.41, 984.71], [918.39, 904.45], [678.25, 446.15], [965.75, 796.15]]) <= 50.0"
] | ballsim | G_3_382 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
144,
147,
222
],
"density": 1,
"position": [
693,
712
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-731.81,
-530.41
]
},
{
"color": [
230,
144,
101
],
"density": 1,
"position": [
748,
656
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-509.77,
-784.74
]
},
{
"color": [
220,
239,
242
],
"density": 1,
"position": [
1014,
772
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-183.86,
858.73
]
},
{
"color": [
123,
131,
82
],
"density": 1,
"position": [
619,
592
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
657.54,
-640.71
]
},
{
"color": [
88,
120,
254
],
"density": 1,
"position": [
601,
676
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-931.3,
251.89
]
},
{
"color": [
223,
230,
210
],
"density": 1,
"position": [
856,
789
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
953.81,
19.61
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 382,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2088054
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (616, 787), initial velocity (-747.53, -528.63) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (799, 747), initial velocity (-816.92, 184.98) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (816, 882), initial velocity (866.53, 210.22) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (652, 697), initial velocity (-776.69, 349.25) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (704, 724), initial velocity (366.3, -740.15) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (736, 654), initial velocity (108.16, -800.43) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[541.24, 734.14], [717.31, 765.51], [902.65, 903.02], [574.34, 731.94], [740.62, 649.98], [746.82, 573.95]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[471.05, 685.85], [635.62, 784.01], [930.94, 865.69], [496.68, 766.87], [777.24, 575.95], [757.63, 493.9]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[524.05, 760.79], [553.73, 802.56], [909.87, 778.83], [454.23, 766.48], [813.95, 501.75], [768.47, 413.65]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[546.64, 742.36], [524.92, 835.62], [888.85, 692.19], [489.15, 688.82], [782.71, 495.6], [720.71, 456.42]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[561.51, 700.15], [607.32, 774.69], [867.83, 605.54], [541.18, 628.25], [746.54, 532.22], [685.31, 536.47]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[576.22, 658.19], [689.72, 713.75], [846.81, 518.9], [584.23, 627.82], [749.15, 568.84], [647.29, 593.94]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[498.78, 775.87], [772.11, 652.81], [787.24, 470.8], [600.68, 586.53], [751.76, 605.47], [686.4, 574.26]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[545.87, 769.01], [854.51, 591.87], [700.59, 449.78], [617.14, 545.25], [754.37, 642.09], [725.52, 554.58]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[663.55, 691.57], [880.72, 587.12], [716.25, 531.08], [648.37, 518.74], [756.99, 678.71], [764.63, 534.91]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[781.23, 614.12], [819.78, 669.52], [737.27, 617.72], [689.65, 502.28], [759.6, 715.33], [803.74, 515.23]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[891.67, 543.92], [770.55, 737.05], [756.96, 700.27], [730.94, 485.83], [742.86, 785.47], [842.85, 495.56]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[814.23, 661.59], [752.68, 830.61], [756.79, 721.7], [772.22, 469.37], [699.3, 902.08], [830.87, 526.98]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[799.35, 716.61], [734.85, 923.93], [716.12, 783.58], [807.94, 458.42], [670.72, 1003.52], [811.24, 565.99]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[827.1, 729.08], [757.0, 948.17], [647.64, 873.33], [791.49, 499.7], [739.18, 1061.75], [791.56, 605.11]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[854.86, 741.56], [805.97, 924.46], [608.09, 934.15], [775.03, 540.98], [794.86, 957.33], [771.89, 644.22]]) <= 50.0"
] | ballsim | G_3_443 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
204,
153,
52
],
"density": 1,
"position": [
616,
787
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-747.53,
-528.63
]
},
{
"color": [
102,
78,
193
],
"density": 1,
"position": [
799,
747
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-816.92,
184.98
]
},
{
"color": [
50,
51,
189
],
"density": 1,
"position": [
816,
882
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
866.53,
210.22
]
},
{
"color": [
111,
60,
93
],
"density": 1,
"position": [
652,
697
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-776.69,
349.25
]
},
{
"color": [
243,
66,
232
],
"density": 1,
"position": [
704,
724
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
366.3,
-740.15
]
},
{
"color": [
244,
80,
193
],
"density": 1,
"position": [
736,
654
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
108.16,
-800.43
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 443,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1244153
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (844, 714), initial velocity (487.66, -728.97) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (702, 857), initial velocity (821.31, -417.85) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (814, 680), initial velocity (207.39, -800.51) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (827, 760), initial velocity (833.36, -479.12) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (686, 789), initial velocity (-188.64, -825.26) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[892.75, 641.1], [784.13, 815.2], [832.58, 603.66], [910.33, 712.1], [667.13, 706.48]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[855.81, 716.66], [866.26, 773.41], [773.62, 661.64], [967.69, 709.17], [648.26, 623.96]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[816.97, 795.55], [948.59, 731.5], [714.52, 719.77], [967.86, 805.53], [629.34, 541.24]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[778.22, 874.23], [1006.08, 732.4], [655.56, 777.76], [897.86, 780.09], [595.89, 483.96]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[736.5, 947.75], [983.96, 777.65], [596.6, 835.74], [814.7, 731.88], [642.41, 508.89]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[648.97, 941.97], [906.72, 727.41], [627.38, 893.72], [731.55, 683.66], [717.36, 544.12]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[603.57, 982.16], [829.47, 677.16], [686.34, 891.97], [648.39, 635.44], [779.4, 601.71]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[643.96, 950.2], [781.45, 716.85], [745.3, 877.88], [599.78, 587.22], [792.61, 603.33]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[644.69, 849.56], [749.2, 805.09], [804.26, 863.8], [682.93, 539.0], [731.08, 658.53]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[645.42, 748.93], [716.95, 893.32], [863.22, 849.71], [703.74, 598.79], [669.56, 713.73]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[598.22, 795.78], [684.7, 981.55], [922.18, 835.62], [703.56, 694.91], [660.42, 607.77]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[615.53, 847.34], [592.19, 965.45], [973.79, 808.8], [703.39, 791.03], [650.0, 501.76]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[664.14, 898.78], [657.51, 949.31], [1015.36, 764.89], [703.22, 886.91], [607.08, 550.71]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[652.85, 929.51], [701.78, 931.04], [1048.45, 735.71], [787.04, 923.65], [707.03, 599.78]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[634.4, 952.31], [657.25, 871.51], [1022.82, 779.34], [867.43, 814.83], [806.98, 648.86]]) <= 50.0"
] | ballsim | G_3_1089 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
180,
171,
131
],
"density": 1,
"position": [
844,
714
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
487.66,
-728.97
]
},
{
"color": [
162,
170,
162
],
"density": 1,
"position": [
702,
857
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
821.31,
-417.85
]
},
{
"color": [
71,
146,
124
],
"density": 1,
"position": [
814,
680
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
207.39,
-800.51
]
},
{
"color": [
164,
131,
172
],
"density": 1,
"position": [
827,
760
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
833.36,
-479.12
]
},
{
"color": [
61,
169,
240
],
"density": 1,
"position": [
686,
789
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-188.64,
-825.26
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1089,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7648201
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (684, 558), initial velocity (818.37, 104.07) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (843, 525), initial velocity (904.7, -180.3) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (534, 551), initial velocity (951.41, -272.38) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (765, 722), initial velocity (-908.0, 29.24) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (620, 826), initial velocity (935.14, 149.34) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (559, 739), initial velocity (720.97, 654.19) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[765.84, 568.4], [933.48, 506.96], [629.14, 523.75], [674.2, 724.93], [713.51, 840.94], [631.09, 804.43]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[847.67, 578.8], [919.42, 549.29], [724.28, 496.51], [583.41, 727.86], [807.01, 855.88], [703.19, 869.86]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[859.28, 659.66], [928.86, 548.36], [819.66, 469.21], [492.39, 730.8], [900.75, 870.86], [775.46, 935.45]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[840.31, 770.87], [968.83, 516.89], [914.8, 443.16], [405.82, 736.17], [994.26, 885.8], [847.56, 1000.88]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[821.34, 882.07], [939.35, 549.79], [898.52, 516.04], [448.69, 816.27], [983.05, 840.29], [919.65, 1045.14]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[802.37, 993.28], [948.84, 625.38], [830.87, 568.38], [491.56, 896.37], [923.36, 766.78], [962.66, 962.93]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[783.4, 1005.0], [958.34, 700.96], [763.22, 620.73], [534.43, 976.47], [863.67, 693.27], [983.26, 867.79]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[764.43, 893.8], [959.24, 767.95], [695.57, 673.07], [577.3, 1056.57], [803.98, 619.76], [1011.1, 779.87]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[745.46, 782.59], [882.05, 756.85], [627.92, 725.41], [620.17, 978.2], [744.28, 546.24], [1098.53, 754.12]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[726.49, 671.38], [804.85, 745.74], [560.27, 777.76], [663.04, 898.1], [684.59, 472.73], [1045.78, 676.57]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[707.52, 560.18], [727.65, 734.63], [492.62, 830.1], [705.91, 817.99], [624.9, 489.34], [979.76, 606.67]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[688.55, 448.96], [650.46, 723.52], [494.74, 842.16], [748.79, 737.89], [565.21, 562.85], [913.75, 536.77]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[669.63, 552.47], [573.45, 712.44], [573.69, 809.82], [791.55, 657.99], [505.66, 636.18], [847.9, 467.05]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[650.66, 663.68], [496.25, 701.33], [652.84, 777.4], [834.42, 577.89], [445.97, 709.69], [781.88, 491.39]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[631.69, 774.88], [434.76, 699.3], [731.99, 744.97], [877.29, 497.78], [432.58, 756.47], [715.87, 561.29]]) <= 50.0"
] | ballsim | G_3_1038 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
208,
226,
185
],
"density": 1,
"position": [
684,
558
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
818.37,
104.07
]
},
{
"color": [
188,
231,
205
],
"density": 1,
"position": [
843,
525
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
904.7,
-180.3
]
},
{
"color": [
163,
190,
88
],
"density": 1,
"position": [
534,
551
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
951.41,
-272.38
]
},
{
"color": [
78,
252,
222
],
"density": 1,
"position": [
765,
722
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-908,
29.24
]
},
{
"color": [
137,
119,
226
],
"density": 1,
"position": [
620,
826
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
935.14,
149.34
]
},
{
"color": [
237,
88,
252
],
"density": 1,
"position": [
559,
739
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
720.97,
654.19
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1038,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8600841
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (573, 522), initial velocity (755.03, -398.9) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (625, 666), initial velocity (661.1, 544.82) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (744, 842), initial velocity (233.91, -912.62) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (781, 593), initial velocity (-223.18, -850.84) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (719, 700), initial velocity (879.24, 334.86) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (867, 937), initial velocity (837.99, -21.04) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[648.51, 482.11], [691.11, 720.49], [767.39, 750.74], [758.69, 507.92], [806.92, 733.5], [950.79, 934.9]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[710.11, 448.29], [757.23, 774.98], [790.78, 659.48], [757.79, 462.31], [894.83, 766.99], [981.88, 902.37]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[708.57, 488.28], [823.51, 829.61], [814.22, 567.99], [854.32, 547.6], [982.96, 800.57], [941.69, 828.55]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[707.03, 528.17], [889.62, 884.1], [837.61, 476.73], [950.61, 632.69], [1049.24, 821.56], [901.6, 754.92]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[705.49, 568.06], [955.73, 938.6], [861.0, 503.07], [1046.89, 717.77], [976.27, 762.16], [861.51, 681.29]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[703.96, 607.96], [936.04, 943.55], [884.39, 594.33], [1040.12, 743.35], [903.3, 702.76], [821.43, 607.65]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[702.42, 647.85], [855.79, 913.55], [898.63, 678.94], [935.05, 714.68], [830.32, 643.37], [781.34, 534.02]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[700.88, 687.74], [775.54, 883.54], [810.47, 689.15], [1017.4, 822.17], [757.35, 583.97], [741.25, 460.4]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[699.34, 727.63], [695.29, 853.54], [722.31, 699.35], [952.01, 844.36], [681.58, 533.18], [703.52, 494.52]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[668.67, 796.66], [615.04, 823.53], [658.67, 685.06], [817.73, 826.78], [569.57, 593.97], [696.29, 467.03]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[636.19, 867.49], [534.79, 793.53], [596.53, 669.27], [683.45, 809.21], [457.55, 654.76], [689.07, 449.02]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[603.72, 938.32], [478.25, 787.22], [534.4, 653.47], [549.17, 791.63], [418.8, 744.86], [681.84, 476.51]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[571.33, 1008.96], [476.44, 877.06], [472.41, 637.71], [492.13, 729.71], [461.4, 837.2], [674.63, 503.93]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[608.68, 1039.46], [600.66, 924.25], [513.18, 657.28], [444.33, 684.22], [510.78, 923.21], [667.4, 531.42]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[686.25, 1046.74], [728.18, 974.74], [583.03, 659.75], [463.77, 739.56], [555.89, 1004.95], [660.18, 558.91]]) <= 50.0"
] | ballsim | G_3_34 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
215,
220,
135
],
"density": 1,
"position": [
573,
522
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
755.03,
-398.9
]
},
{
"color": [
110,
99,
58
],
"density": 1,
"position": [
625,
666
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
661.1,
544.82
]
},
{
"color": [
157,
109,
141
],
"density": 1,
"position": [
744,
842
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
233.91,
-912.62
]
},
{
"color": [
51,
141,
158
],
"density": 1,
"position": [
781,
593
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-223.18,
-850.84
]
},
{
"color": [
100,
79,
231
],
"density": 1,
"position": [
719,
700
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
879.24,
334.86
]
},
{
"color": [
206,
174,
108
],
"density": 1,
"position": [
867,
937
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
837.99,
-21.04
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 34,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1083629
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (761, 770), initial velocity (877.45, -276.96) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (676, 808), initial velocity (-892.59, -192.46) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (923, 786), initial velocity (-420.9, 740.84) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (1022, 666), initial velocity (-866.23, 19.23) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (619, 921), initial velocity (430.45, 793.53) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[848.74, 742.31], [586.74, 788.76], [880.91, 860.07], [935.38, 667.93], [662.04, 1000.35]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[936.49, 714.63], [497.48, 769.52], [838.82, 934.14], [848.76, 669.86], [705.08, 1029.81]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[1024.45, 686.87], [408.0, 750.24], [796.63, 1008.4], [761.92, 671.79], [748.23, 950.26]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1022.72, 710.83], [449.6, 806.54], [754.54, 1027.02], [675.3, 673.72], [791.28, 870.92]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[954.87, 772.97], [510.89, 874.22], [712.45, 952.95], [588.68, 675.65], [834.32, 791.57]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[887.02, 835.1], [572.2, 941.89], [670.36, 878.88], [502.06, 677.58], [877.36, 712.23]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[819.18, 897.24], [633.5, 1009.57], [628.27, 804.81], [445.85, 697.06], [920.4, 632.88]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[751.33, 959.37], [694.8, 1037.61], [586.18, 730.74], [487.5, 773.04], [963.44, 553.54]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[726.12, 947.65], [690.44, 1031.21], [544.09, 656.66], [529.15, 849.02], [915.38, 526.79]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[727.95, 889.08], [644.42, 913.02], [502.0, 582.59], [570.8, 924.99], [825.15, 524.4]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[729.78, 830.51], [598.4, 794.83], [571.7, 573.06], [612.45, 1000.97], [734.91, 522.01]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[731.61, 771.94], [552.38, 676.64], [656.91, 572.47], [654.1, 1037.91], [644.68, 519.61]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[733.44, 713.52], [506.47, 558.75], [741.9, 571.89], [695.65, 962.12], [554.67, 517.23]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[735.27, 654.95], [575.9, 632.76], [827.1, 571.3], [737.3, 886.15], [599.26, 511.15]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[737.1, 596.38], [627.05, 742.03], [912.31, 570.72], [778.95, 810.17], [694.65, 504.63]]) <= 50.0"
] | ballsim | G_3_13 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
186,
171,
183
],
"density": 1,
"position": [
761,
770
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
877.45,
-276.96
]
},
{
"color": [
205,
199,
68
],
"density": 1,
"position": [
676,
808
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-892.59,
-192.46
]
},
{
"color": [
149,
131,
85
],
"density": 1,
"position": [
923,
786
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-420.9,
740.84
]
},
{
"color": [
117,
208,
98
],
"density": 1,
"position": [
1022,
666
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-866.23,
19.23
]
},
{
"color": [
98,
166,
165
],
"density": 1,
"position": [
619,
921
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
430.45,
793.53
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 13,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5468731
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (921, 714), initial velocity (815.95, 78.92) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (801, 472), initial velocity (876.43, 145.28) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (876, 686), initial velocity (-931.68, 114.08) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (797, 566), initial velocity (412.93, 796.62) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (922, 567), initial velocity (272.57, 802.81) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[1002.59, 721.89], [888.65, 486.53], [782.84, 697.4], [838.28, 645.66], [949.25, 647.27]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[1084.18, 729.77], [953.74, 514.07], [689.67, 708.8], [879.57, 725.33], [976.49, 727.55]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[1063.81, 796.66], [922.41, 597.43], [596.27, 720.23], [920.96, 805.19], [1003.81, 808.02]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1022.17, 836.51], [891.16, 680.59], [503.11, 731.63], [962.24, 884.85], [1007.51, 889.87]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[959.96, 817.4], [859.91, 763.74], [409.95, 743.04], [939.55, 917.29], [946.77, 963.2]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[897.75, 798.28], [828.66, 846.9], [437.99, 815.53], [896.7, 924.25], [862.86, 1030.06]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[835.55, 779.16], [797.41, 930.05], [494.46, 890.5], [853.85, 931.2], [778.94, 1017.95]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[773.34, 760.05], [766.16, 1013.21], [550.93, 965.48], [811.01, 938.16], [695.03, 951.08]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[711.13, 740.93], [734.91, 1015.1], [607.4, 1040.46], [768.16, 945.12], [611.12, 884.21]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[648.92, 721.82], [687.21, 943.89], [663.87, 999.41], [755.44, 930.19], [527.21, 817.34]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[586.72, 702.7], [620.31, 886.61], [720.34, 924.43], [777.85, 889.74], [443.29, 750.47]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[524.51, 683.58], [553.42, 829.34], [772.44, 849.46], [809.0, 849.28], [470.34, 747.66]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[462.47, 664.52], [511.02, 847.06], [806.12, 774.67], [876.66, 808.93], [547.7, 718.31]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[482.98, 693.15], [509.52, 921.09], [839.88, 699.69], [944.48, 768.48], [605.24, 627.29]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[530.64, 737.46], [593.53, 945.75], [873.65, 624.72], [1012.3, 728.02], [662.79, 536.28]]) <= 50.0"
] | ballsim | G_3_99 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
101,
109,
113
],
"density": 1,
"position": [
921,
714
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
815.95,
78.92
]
},
{
"color": [
98,
199,
250
],
"density": 1,
"position": [
801,
472
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
876.43,
145.28
]
},
{
"color": [
244,
255,
228
],
"density": 1,
"position": [
876,
686
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-931.68,
114.08
]
},
{
"color": [
126,
209,
186
],
"density": 1,
"position": [
797,
566
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
412.93,
796.62
]
},
{
"color": [
192,
81,
162
],
"density": 1,
"position": [
922,
567
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
272.57,
802.81
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 99,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2247798
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (713, 802), initial velocity (-563.94, -614.71) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (877, 711), initial velocity (-653.93, 580.49) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (592, 656), initial velocity (-826.73, -540.12) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (657, 746), initial velocity (117.72, -873.21) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (814, 677), initial velocity (-502.48, -863.44) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[656.6, 740.53], [811.62, 769.06], [509.33, 602.0], [668.77, 658.67], [763.76, 590.65]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[600.21, 679.05], [746.24, 827.11], [506.6, 547.99], [680.54, 571.34], [713.51, 504.3]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[543.67, 617.42], [680.69, 885.32], [595.29, 511.74], [653.21, 522.92], [738.42, 489.87]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[487.27, 555.95], [615.31, 943.37], [626.21, 543.01], [697.85, 603.14], [842.72, 495.07]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[466.66, 604.47], [560.52, 968.83], [655.49, 608.93], [763.13, 695.28], [916.29, 522.58]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[476.16, 687.35], [541.75, 883.43], [684.76, 674.84], [828.41, 787.42], [889.0, 623.37]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[485.65, 770.24], [522.97, 798.03], [714.03, 740.76], [893.7, 879.56], [861.7, 724.15]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[481.17, 853.12], [532.15, 712.63], [743.3, 806.68], [939.26, 957.37], [834.41, 824.93]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[471.8, 936.01], [551.07, 627.23], [772.58, 872.6], [840.25, 932.58], [798.31, 916.9]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[500.01, 903.27], [569.99, 541.83], [801.85, 881.45], [839.91, 1006.44], [663.57, 984.39]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[541.14, 830.71], [616.83, 542.35], [831.12, 884.23], [840.86, 1076.32], [577.39, 910.28]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[582.28, 758.15], [682.33, 600.31], [860.39, 887.02], [884.0, 1016.36], [512.67, 770.11]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[623.32, 685.78], [747.67, 658.12], [889.59, 889.79], [927.03, 956.55], [485.14, 630.3]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[664.45, 613.22], [813.17, 716.08], [918.87, 892.58], [970.17, 896.59], [565.08, 537.02]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[705.59, 540.66], [878.68, 774.04], [948.14, 895.36], [1013.31, 836.63], [699.83, 612.39]]) <= 50.0"
] | ballsim | G_3_1050 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
118,
89,
64
],
"density": 1,
"position": [
713,
802
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-563.94,
-614.71
]
},
{
"color": [
138,
85,
253
],
"density": 1,
"position": [
877,
711
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-653.93,
580.49
]
},
{
"color": [
116,
70,
85
],
"density": 1,
"position": [
592,
656
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-826.73,
-540.12
]
},
{
"color": [
147,
157,
124
],
"density": 1,
"position": [
657,
746
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
117.72,
-873.21
]
},
{
"color": [
189,
189,
152
],
"density": 1,
"position": [
814,
677
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-502.48,
-863.44
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1050,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8649776
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (813, 593), initial velocity (43.91, -830.58) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (533, 756), initial velocity (355.45, -808.82) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (878, 728), initial velocity (954.43, -136.06) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (911, 804), initial velocity (-992.43, -111.05) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (592, 776), initial velocity (-760.16, -581.49) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[817.39, 509.94], [536.24, 642.81], [973.43, 714.4], [811.76, 792.89], [540.85, 742.72]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[781.87, 466.8], [633.28, 623.42], [1050.58, 719.09], [712.51, 781.78], [503.47, 723.21]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[698.61, 471.21], [764.69, 638.11], [1035.02, 812.86], [613.02, 770.65], [466.0, 703.66]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[676.73, 536.78], [895.77, 652.75], [939.58, 826.46], [513.78, 759.54], [468.19, 723.71]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[672.34, 619.84], [1022.39, 671.86], [844.15, 840.06], [511.23, 804.25], [442.32, 753.12]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[667.94, 702.9], [1037.04, 802.94], [748.71, 853.65], [526.93, 859.5], [491.12, 813.34]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[663.55, 785.95], [920.97, 803.3], [653.28, 867.25], [567.7, 889.68], [576.79, 836.68]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[659.15, 869.01], [789.89, 788.65], [664.02, 942.16], [622.95, 905.37], [565.29, 803.92]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[729.72, 897.61], [658.81, 774.0], [714.82, 1040.21], [637.24, 950.82], [517.12, 749.99]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[817.25, 913.89], [527.73, 759.36], [737.1, 1058.58], [655.94, 989.32], [468.94, 696.06]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[904.77, 930.18], [400.31, 748.35], [745.94, 1003.87], [648.47, 951.01], [510.21, 731.56]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[908.52, 862.69], [524.43, 769.94], [811.57, 990.42], [584.21, 871.45], [564.14, 779.73]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[892.28, 775.38], [610.86, 758.97], [872.79, 972.75], [520.11, 792.08], [652.05, 847.49]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[876.0, 687.86], [692.97, 745.34], [886.24, 907.13], [455.87, 712.51], [743.68, 917.43]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[859.71, 600.34], [775.07, 731.72], [899.69, 841.5], [520.0, 761.34], [835.3, 987.38]]) <= 50.0"
] | ballsim | G_3_905 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
182,
100,
248
],
"density": 1,
"position": [
813,
593
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
43.91,
-830.58
]
},
{
"color": [
51,
87,
170
],
"density": 1,
"position": [
533,
756
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
355.45,
-808.82
]
},
{
"color": [
62,
161,
114
],
"density": 1,
"position": [
878,
728
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
954.43,
-136.06
]
},
{
"color": [
214,
112,
170
],
"density": 1,
"position": [
911,
804
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-992.43,
-111.05
]
},
{
"color": [
203,
242,
240
],
"density": 1,
"position": [
592,
776
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-760.16,
-581.49
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 905,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 502064
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (701, 813), initial velocity (-688.68, -499.96) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (680, 760), initial velocity (-646.22, -571.37) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (860, 718), initial velocity (864.98, -251.08) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (790, 698), initial velocity (605.45, 674.94) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (783, 821), initial velocity (953.51, -81.75) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[632.13, 763.0], [615.38, 702.87], [946.5, 692.9], [850.55, 765.5], [878.36, 812.82]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[581.76, 713.0], [594.27, 645.74], [988.24, 745.33], [911.09, 833.01], [973.72, 804.64]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[650.81, 662.88], [659.05, 588.47], [1037.27, 748.48], [895.98, 772.69], [928.87, 788.38]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[719.68, 612.88], [723.68, 531.34], [1045.19, 757.82], [807.87, 686.51], [930.65, 768.26]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[784.71, 569.53], [684.1, 572.58], [981.55, 722.81], [731.61, 620.85], [932.44, 748.14]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[775.85, 654.17], [588.87, 465.22], [917.9, 687.8], [704.58, 640.51], [934.22, 728.02]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[766.99, 738.82], [596.58, 649.5], [854.25, 652.79], [677.56, 660.16], [936.0, 707.91]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[758.12, 823.46], [637.01, 833.78], [790.61, 617.78], [650.53, 679.81], [937.78, 687.79]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[749.26, 908.1], [659.06, 986.24], [726.96, 582.77], [623.5, 699.47], [939.56, 667.67]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[712.47, 944.36], [625.34, 859.09], [663.31, 547.76], [596.48, 719.12], [934.43, 659.54]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[634.73, 909.72], [764.72, 731.94], [599.66, 512.75], [595.57, 738.77], [917.91, 671.16]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[588.02, 875.08], [878.78, 648.63], [617.83, 483.87], [622.59, 758.43], [901.38, 682.78]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[665.56, 840.52], [775.28, 722.99], [619.31, 556.31], [649.55, 778.03], [948.09, 803.81]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[743.3, 805.88], [711.06, 762.13], [620.8, 628.94], [654.09, 810.67], [892.82, 748.32]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[821.03, 771.23], [764.33, 733.45], [622.29, 701.58], [599.86, 877.22], [805.71, 637.68]]) <= 50.0"
] | ballsim | G_3_884 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
63,
74,
111
],
"density": 1,
"position": [
701,
813
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-688.68,
-499.96
]
},
{
"color": [
54,
235,
132
],
"density": 1,
"position": [
680,
760
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-646.22,
-571.37
]
},
{
"color": [
219,
89,
237
],
"density": 1,
"position": [
860,
718
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
864.98,
-251.08
]
},
{
"color": [
110,
70,
251
],
"density": 1,
"position": [
790,
698
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
605.45,
674.94
]
},
{
"color": [
63,
82,
98
],
"density": 1,
"position": [
783,
821
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
953.51,
-81.75
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 884,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8246781
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (562, 901), initial velocity (721.81, 649.79) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (931, 832), initial velocity (-128.94, -890.54) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (868, 574), initial velocity (-934.2, 156.89) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (656, 618), initial velocity (-946.67, -104.43) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (798, 838), initial velocity (619.05, 671.65) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[634.19, 965.99], [918.11, 742.94], [774.59, 589.7], [561.32, 607.55], [859.91, 905.16]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[706.38, 1030.98], [905.22, 653.88], [681.18, 605.4], [466.65, 597.1], [921.83, 972.33]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[801.29, 1026.7], [892.3, 564.59], [587.54, 621.13], [541.51, 586.63], [863.54, 952.23]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[897.87, 1016.56], [879.41, 475.53], [494.13, 636.83], [636.19, 576.18], [780.54, 914.12]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[895.28, 934.35], [838.94, 429.34], [532.54, 652.53], [730.87, 565.73], [697.53, 876.01]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[875.09, 839.36], [797.02, 508.97], [625.95, 668.23], [825.55, 555.28], [614.52, 837.9]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[854.9, 744.36], [755.11, 588.58], [719.36, 683.93], [920.22, 544.83], [531.51, 799.79]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[834.71, 649.37], [713.19, 668.2], [812.77, 699.63], [927.77, 597.69], [484.75, 761.68]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[814.52, 554.37], [671.27, 747.81], [906.18, 715.32], [888.58, 684.51], [567.76, 723.57]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[794.33, 459.38], [629.35, 827.42], [999.58, 731.02], [849.4, 771.33], [650.77, 685.46]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[820.78, 507.96], [587.43, 907.04], [1092.99, 746.72], [810.21, 858.14], [733.78, 647.35]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[860.29, 596.68], [546.17, 984.62], [1031.81, 787.81], [771.03, 944.96], [816.78, 609.24]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[899.69, 685.18], [559.03, 895.78], [947.23, 829.92], [731.94, 1031.56], [899.58, 571.22]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[939.19, 773.9], [571.92, 806.72], [862.44, 872.13], [744.8, 958.17], [933.15, 569.03]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[978.69, 862.62], [584.81, 717.65], [777.65, 914.34], [764.11, 864.91], [871.24, 636.19]]) <= 50.0"
] | ballsim | G_3_590 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
198,
104,
229
],
"density": 1,
"position": [
562,
901
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
721.81,
649.79
]
},
{
"color": [
243,
213,
115
],
"density": 1,
"position": [
931,
832
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-128.94,
-890.54
]
},
{
"color": [
67,
222,
93
],
"density": 1,
"position": [
868,
574
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-934.2,
156.89
]
},
{
"color": [
182,
136,
195
],
"density": 1,
"position": [
656,
618
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-946.67,
-104.43
]
},
{
"color": [
115,
157,
156
],
"density": 1,
"position": [
798,
838
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
619.05,
671.65
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 590,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7087707
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (739, 970), initial velocity (742.46, -602.36) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (924, 752), initial velocity (609.03, -553.53) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (685, 888), initial velocity (-737.9, 440.41) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (963, 794), initial velocity (-314.49, 880.59) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (930, 521), initial velocity (-643.09, 694.86) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (761, 749), initial velocity (-602.36, -724.73) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[813.24, 909.77], [984.91, 696.65], [611.22, 932.04], [931.55, 882.06], [865.69, 590.48], [700.77, 676.54]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[887.49, 849.54], [1040.44, 644.41], [537.44, 976.09], [900.11, 970.12], [801.39, 659.96], [640.54, 604.08]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[961.92, 789.16], [961.86, 669.55], [594.81, 944.39], [868.59, 1053.06], [736.92, 729.62], [580.16, 531.44]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1036.16, 728.93], [883.47, 694.62], [669.83, 902.5], [837.14, 965.01], [672.61, 799.1], [589.04, 498.87]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[1031.86, 714.05], [805.07, 719.69], [744.85, 860.61], [805.69, 876.95], [608.31, 868.59], [681.91, 514.8]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[942.57, 748.23], [726.68, 744.77], [741.68, 818.71], [859.71, 788.89], [544.0, 938.07], [774.78, 530.72]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[1003.56, 782.41], [648.28, 769.84], [714.95, 776.82], [857.36, 700.82], [575.72, 952.11], [867.66, 546.64]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[1033.35, 759.12], [569.89, 794.91], [688.21, 734.92], [817.66, 612.76], [668.05, 931.17], [960.53, 562.55]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[939.09, 664.2], [491.5, 819.99], [661.48, 693.03], [777.97, 524.7], [760.39, 910.22], [982.24, 619.57]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[844.83, 569.28], [468.28, 813.2], [634.75, 651.13], [738.27, 451.9], [852.72, 889.27], [949.6, 707.97]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[750.56, 474.35], [529.19, 757.85], [608.01, 609.24], [698.57, 539.96], [945.06, 868.32], [916.96, 796.37]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[656.3, 505.71], [590.1, 702.51], [581.28, 567.34], [658.87, 628.03], [1037.39, 847.38], [884.32, 884.78]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[562.27, 600.39], [570.42, 727.73], [554.61, 525.55], [686.93, 648.21], [1026.5, 767.03], [851.76, 972.96]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[468.01, 695.32], [498.79, 804.9], [559.23, 501.77], [758.71, 624.8], [998.47, 676.6], [819.11, 1050.11]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[440.5, 751.69], [493.32, 843.88], [608.86, 503.98], [830.49, 601.38], [970.45, 586.17], [786.47, 961.71]]) <= 50.0"
] | ballsim | G_3_150 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
113,
104,
204
],
"density": 1,
"position": [
739,
970
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
742.46,
-602.36
]
},
{
"color": [
230,
70,
60
],
"density": 1,
"position": [
924,
752
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
609.03,
-553.53
]
},
{
"color": [
197,
142,
169
],
"density": 1,
"position": [
685,
888
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-737.9,
440.41
]
},
{
"color": [
167,
204,
176
],
"density": 1,
"position": [
963,
794
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-314.49,
880.59
]
},
{
"color": [
91,
206,
50
],
"density": 1,
"position": [
930,
521
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-643.09,
694.86
]
},
{
"color": [
187,
88,
70
],
"density": 1,
"position": [
761,
749
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-602.36,
-724.73
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 150,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8524141
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (598, 595), initial velocity (-764.21, -459.12) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (738, 850), initial velocity (956.04, -125.3) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (740, 698), initial velocity (500.06, 635.42) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (736, 757), initial velocity (-827.14, 228.17) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (805, 827), initial velocity (-197.8, 953.23) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (780, 643), initial velocity (-855.92, -323.65) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[521.58, 549.08], [804.04, 820.41], [790.0, 761.55], [653.29, 779.83], [814.79, 939.38], [694.4, 610.63]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[470.98, 582.6], [878.12, 816.61], [803.38, 761.66], [570.57, 802.65], [834.85, 1057.69], [608.81, 578.25]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[489.57, 664.86], [961.72, 828.99], [798.09, 729.38], [487.65, 825.54], [910.08, 1006.65], [523.0, 545.8]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[524.4, 746.92], [1033.23, 832.71], [792.82, 697.18], [520.68, 848.37], [971.07, 904.7], [466.95, 578.85]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[559.24, 828.97], [995.71, 757.19], [787.55, 664.98], [603.4, 871.19], [1024.39, 797.18], [517.17, 655.34]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[594.08, 911.03], [958.19, 681.68], [782.27, 632.77], [686.11, 894.02], [1052.65, 707.87], [567.39, 731.83]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[628.92, 993.08], [920.66, 606.17], [777.0, 600.57], [768.83, 916.85], [933.92, 725.35], [617.61, 808.32]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[693.85, 982.52], [883.14, 530.65], [771.73, 568.37], [851.54, 939.67], [815.19, 742.83], [667.83, 884.81]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[770.26, 936.6], [845.62, 455.14], [766.45, 536.17], [934.26, 962.5], [696.47, 760.31], [718.05, 961.3]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[846.68, 890.68], [839.3, 475.65], [761.18, 503.97], [908.38, 906.43], [577.74, 777.79], [768.27, 1037.79]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[824.92, 844.76], [853.34, 558.79], [755.91, 471.76], [959.29, 834.83], [474.24, 795.27], [842.04, 1041.8]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[777.66, 798.83], [867.37, 641.92], [757.58, 460.94], [1035.7, 763.22], [592.97, 812.75], [913.06, 998.84]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[730.51, 753.02], [881.38, 724.84], [772.22, 490.02], [1047.47, 738.62], [711.4, 830.19], [917.37, 907.66]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[683.24, 707.1], [886.87, 803.04], [786.89, 519.17], [955.74, 789.18], [830.12, 847.67], [931.03, 821.64]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[635.98, 661.18], [821.71, 840.46], [801.57, 548.32], [907.78, 705.06], [948.85, 865.15], [978.13, 914.87]]) <= 50.0"
] | ballsim | G_3_1039 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
209,
114,
194
],
"density": 1,
"position": [
598,
595
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-764.21,
-459.12
]
},
{
"color": [
182,
84,
209
],
"density": 1,
"position": [
738,
850
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
956.04,
-125.3
]
},
{
"color": [
182,
234,
93
],
"density": 1,
"position": [
740,
698
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
500.06,
635.42
]
},
{
"color": [
191,
88,
71
],
"density": 1,
"position": [
736,
757
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-827.14,
228.17
]
},
{
"color": [
186,
164,
123
],
"density": 1,
"position": [
805,
827
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-197.8,
953.23
]
},
{
"color": [
66,
196,
237
],
"density": 1,
"position": [
780,
643
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-855.92,
-323.65
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 1039,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2709234
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (776, 828), initial velocity (-891.28, -293.03) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (739, 683), initial velocity (-823.01, -352.49) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (538, 937), initial velocity (341.02, 881.09) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (1007, 713), initial velocity (979.07, 42.92) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (860, 809), initial velocity (-77.27, 912.38) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[686.86, 798.7], [656.7, 647.75], [572.11, 1025.11], [1076.71, 733.57], [852.29, 900.24]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[597.73, 769.41], [574.4, 612.49], [606.21, 1001.66], [1031.5, 820.51], [844.57, 991.47]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[508.37, 740.04], [500.71, 582.24], [640.4, 913.33], [986.15, 907.67], [836.84, 1028.51]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[419.25, 710.74], [572.39, 635.9], [674.51, 825.21], [940.91, 994.61], [829.12, 937.27]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[486.84, 771.91], [644.07, 689.57], [708.62, 737.1], [895.67, 1027.97], [821.41, 846.04]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[556.79, 834.43], [715.75, 743.23], [742.72, 648.99], [850.43, 941.03], [813.69, 754.8]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[626.73, 896.95], [787.43, 796.89], [776.83, 560.88], [805.19, 854.09], [805.98, 663.56]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[696.68, 959.48], [882.78, 777.65], [810.94, 472.77], [731.81, 853.81], [798.26, 572.33]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[766.63, 1022.0], [982.0, 746.5], [845.04, 500.48], [653.83, 867.67], [790.55, 481.09]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[836.57, 1024.95], [1081.22, 715.34], [879.15, 588.59], [575.86, 881.54], [782.83, 498.68]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[906.52, 962.42], [1010.25, 782.45], [913.26, 676.7], [497.87, 895.41], [775.12, 589.92]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[976.46, 899.9], [933.67, 852.81], [947.36, 764.81], [520.51, 851.19], [767.4, 681.15]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[1046.23, 837.53], [857.27, 923.0], [981.38, 852.7], [571.39, 790.74], [759.71, 772.16]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[1077.75, 752.81], [780.68, 993.36], [984.49, 922.92], [622.39, 730.15], [751.99, 863.4]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[1031.92, 698.52], [704.1, 1047.74], [891.13, 937.44], [673.39, 669.55], [744.28, 954.63]]) <= 50.0"
] | ballsim | G_3_835 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
126,
211,
55
],
"density": 1,
"position": [
776,
828
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-891.28,
-293.03
]
},
{
"color": [
232,
247,
174
],
"density": 1,
"position": [
739,
683
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-823.01,
-352.49
]
},
{
"color": [
252,
147,
215
],
"density": 1,
"position": [
538,
937
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
341.02,
881.09
]
},
{
"color": [
171,
74,
221
],
"density": 1,
"position": [
1007,
713
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
979.07,
42.92
]
},
{
"color": [
131,
232,
239
],
"density": 1,
"position": [
860,
809
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-77.27,
912.38
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 835,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5587552
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (608, 756), initial velocity (78.37, -822.02) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (883, 899), initial velocity (-682.59, -695.06) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (675, 524), initial velocity (808.48, -123.12) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (853, 822), initial velocity (857.2, 283.39) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (665, 748), initial velocity (395.61, -863.72) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[615.84, 673.8], [814.74, 829.49], [755.86, 511.7], [938.72, 850.34], [704.55, 661.62]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[623.67, 591.6], [746.48, 759.99], [836.72, 499.38], [976.3, 830.55], [744.1, 575.25]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[643.76, 521.42], [678.04, 690.31], [842.03, 562.79], [947.88, 744.62], [783.75, 488.65]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[725.96, 513.58], [609.78, 620.8], [829.72, 643.65], [919.54, 658.9], [757.26, 468.32]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[723.81, 590.11], [601.04, 610.81], [817.42, 724.51], [891.19, 573.18], [762.59, 416.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[707.33, 680.95], [670.55, 679.08], [805.11, 805.37], [842.45, 507.86], [697.9, 460.17]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[751.2, 771.8], [629.61, 747.34], [792.81, 886.23], [756.73, 479.52], [666.96, 537.91]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[795.49, 862.64], [587.91, 815.6], [780.5, 967.09], [701.16, 481.32], [636.03, 615.64]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[839.77, 953.49], [546.21, 883.86], [768.2, 1047.95], [729.51, 567.04], [605.1, 693.38]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[805.45, 965.73], [604.46, 852.18], [744.11, 1081.16], [757.85, 652.75], [574.17, 771.11]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[714.61, 921.44], [672.72, 810.48], [756.42, 1000.3], [786.2, 738.47], [543.23, 848.85]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[623.76, 877.15], [740.98, 768.78], [768.72, 919.44], [814.54, 824.19], [571.59, 867.29]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[672.21, 832.97], [809.07, 727.18], [781.0, 838.78], [842.82, 909.69], [483.84, 836.43]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[735.42, 788.69], [877.33, 685.48], [793.3, 757.92], [859.58, 983.83], [450.51, 733.46]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[798.63, 744.4], [945.59, 643.79], [805.6, 677.06], [773.86, 955.48], [516.93, 725.43]]) <= 50.0"
] | ballsim | G_3_80 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
219,
116,
236
],
"density": 1,
"position": [
608,
756
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
78.37,
-822.02
]
},
{
"color": [
218,
139,
255
],
"density": 1,
"position": [
883,
899
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-682.59,
-695.06
]
},
{
"color": [
174,
157,
200
],
"density": 1,
"position": [
675,
524
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
808.48,
-123.12
]
},
{
"color": [
110,
206,
187
],
"density": 1,
"position": [
853,
822
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
857.2,
283.39
]
},
{
"color": [
188,
52,
129
],
"density": 1,
"position": [
665,
748
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
395.61,
-863.72
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 80,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8674702
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (846, 727), initial velocity (613.55, -633.52) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (722, 760), initial velocity (-134.21, 834.51) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (706, 582), initial velocity (588.25, -647.0) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (968, 600), initial velocity (-692.6, -469.63) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (610, 864), initial velocity (695.12, 690.33) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[907.35, 663.65], [708.57, 843.45], [764.81, 517.3], [898.74, 553.03], [679.51, 933.04]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[968.71, 600.29], [695.14, 926.89], [823.63, 452.61], [829.47, 506.06], [749.01, 1002.09]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[925.32, 612.99], [681.68, 1010.55], [861.13, 449.75], [760.04, 458.98], [818.69, 1071.3]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[846.12, 651.76], [710.66, 963.5], [851.56, 536.67], [723.27, 511.97], [872.23, 1024.86]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[766.92, 690.53], [748.84, 888.11], [841.99, 623.58], [694.85, 590.68], [856.58, 928.16]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[687.72, 729.3], [787.03, 812.72], [832.42, 710.5], [666.43, 669.39], [840.93, 831.45]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[606.02, 775.76], [768.65, 738.75], [854.45, 715.79], [640.51, 740.42], [878.22, 785.9]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[519.91, 835.72], [707.32, 670.95], [893.32, 626.34], [618.98, 797.93], [960.37, 793.86]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[491.81, 895.68], [645.99, 603.15], [932.19, 536.89], [597.44, 855.45], [1042.52, 801.81]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[577.91, 955.64], [584.66, 535.36], [858.61, 529.13], [575.91, 912.97], [1042.06, 749.76]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[664.02, 1015.6], [557.24, 571.9], [761.52, 538.45], [554.38, 970.49], [1009.1, 674.1]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[763.97, 1032.96], [547.47, 662.79], [664.42, 547.78], [560.23, 943.72], [976.14, 598.45]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[868.62, 1035.05], [537.73, 753.46], [567.57, 557.08], [576.57, 884.69], [939.05, 526.03]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[872.53, 963.77], [527.97, 844.35], [470.47, 566.41], [592.95, 825.51], [877.26, 471.31]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[838.13, 864.65], [518.2, 935.24], [539.9, 575.73], [609.33, 766.33], [823.2, 440.39]]) <= 50.0"
] | ballsim | G_3_161 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
200,
177,
214
],
"density": 1,
"position": [
846,
727
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
613.55,
-633.52
]
},
{
"color": [
208,
69,
213
],
"density": 1,
"position": [
722,
760
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-134.21,
834.51
]
},
{
"color": [
143,
161,
209
],
"density": 1,
"position": [
706,
582
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
588.25,
-647
]
},
{
"color": [
172,
196,
145
],
"density": 1,
"position": [
968,
600
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-692.6,
-469.63
]
},
{
"color": [
125,
109,
75
],
"density": 1,
"position": [
610,
864
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
695.12,
690.33
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 161,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1967049
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (701, 489), initial velocity (349.56, 827.82) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (685, 742), initial velocity (852.79, -481.39) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (629, 655), initial velocity (-962.91, -123.67) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (982, 810), initial velocity (-542.25, 770.32) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (752, 616), initial velocity (546.57, -713.86) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (762, 704), initial velocity (-870.49, 163.33) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[735.96, 571.79], [666.01, 769.61], [532.71, 642.62], [927.78, 887.03], [806.66, 544.61], [779.21, 644.58]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[770.92, 654.58], [607.83, 825.69], [478.01, 654.26], [873.55, 964.05], [861.33, 473.23], [835.6, 556.69]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[805.97, 737.57], [549.51, 881.91], [537.02, 731.65], [819.19, 1041.28], [938.41, 470.69], [869.86, 484.77]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[840.93, 820.36], [514.1, 924.84], [595.88, 808.85], [764.97, 996.54], [894.07, 587.76], [849.37, 452.74]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[875.89, 903.15], [591.76, 902.5], [654.74, 886.04], [710.75, 919.51], [841.75, 709.44], [828.89, 467.83]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[910.85, 985.94], [669.42, 880.16], [645.01, 963.24], [759.02, 847.84], [791.12, 828.2], [808.41, 499.86]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[908.56, 1047.21], [747.09, 857.82], [614.81, 1040.44], [686.9, 1039.28], [823.49, 803.2], [787.92, 531.9]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[819.4, 1053.11], [824.75, 835.48], [584.61, 991.86], [614.78, 884.15], [855.87, 778.2], [767.44, 563.93]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[730.24, 1041.98], [902.41, 813.14], [554.41, 914.67], [542.66, 692.71], [888.24, 753.2], [746.96, 595.96]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[641.08, 1030.85], [980.07, 790.81], [524.21, 837.47], [572.64, 560.23], [920.61, 728.2], [726.47, 627.99]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[554.59, 1018.18], [1057.72, 768.47], [494.01, 760.27], [774.5, 526.98], [952.98, 703.2], [705.99, 660.02]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[589.56, 935.39], [1086.45, 725.63], [463.81, 683.07], [947.36, 510.46], [985.36, 678.2], [685.51, 692.05]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[624.43, 852.81], [1042.84, 680.15], [500.32, 644.54], [817.95, 668.24], [1003.25, 642.8], [665.07, 724.0]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[665.05, 787.44], [1058.71, 677.86], [582.28, 632.08], [688.21, 826.42], [961.57, 564.01], [638.93, 738.62]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[728.45, 792.23], [1055.74, 686.45], [664.24, 619.63], [558.48, 984.6], [919.9, 485.22], [590.01, 683.1]]) <= 50.0"
] | ballsim | G_3_775 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
135,
181,
118
],
"density": 1,
"position": [
701,
489
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
349.56,
827.82
]
},
{
"color": [
219,
157,
153
],
"density": 1,
"position": [
685,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
852.79,
-481.39
]
},
{
"color": [
165,
245,
202
],
"density": 1,
"position": [
629,
655
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-962.91,
-123.67
]
},
{
"color": [
223,
133,
201
],
"density": 1,
"position": [
982,
810
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-542.25,
770.32
]
},
{
"color": [
222,
195,
67
],
"density": 1,
"position": [
752,
616
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
546.57,
-713.86
]
},
{
"color": [
210,
209,
157
],
"density": 1,
"position": [
762,
704
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-870.49,
163.33
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 775,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1004128
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (503, 753), initial velocity (-973.71, -3.25) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (702, 817), initial velocity (260.06, -830.26) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (755, 704), initial velocity (31.18, 850.81) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (687, 751), initial velocity (685.36, 622.09) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (822, 607), initial velocity (-688.3, 489.24) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (870, 528), initial velocity (109.67, -877.92) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[527.63, 752.68], [766.61, 852.78], [758.12, 789.08], [720.2, 704.48], [753.18, 655.93], [879.58, 441.21]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[624.99, 752.37], [841.44, 919.99], [761.25, 874.17], [706.8, 743.87], [725.07, 579.55], [800.12, 447.36]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[704.21, 733.64], [916.46, 987.37], [764.38, 959.46], [701.44, 816.3], [703.55, 482.45], [739.5, 512.1]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[768.59, 700.32], [850.46, 952.26], [767.51, 1044.54], [707.38, 899.82], [731.74, 538.45], [679.03, 576.68]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[832.97, 666.99], [763.43, 901.87], [814.42, 994.85], [713.31, 983.34], [771.29, 629.42], [618.55, 641.25]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[897.35, 633.67], [676.39, 851.48], [866.96, 927.86], [735.96, 1015.4], [810.84, 720.38], [558.08, 705.83]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[961.73, 600.34], [589.36, 801.09], [919.5, 860.87], [789.84, 951.31], [850.39, 811.35], [497.6, 770.4]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[955.34, 618.43], [502.32, 750.7], [972.04, 793.87], [843.72, 887.22], [889.95, 902.32], [496.13, 834.98]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[903.75, 669.36], [510.35, 700.31], [1024.58, 726.88], [897.61, 823.14], [927.84, 992.08], [556.6, 899.55]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[852.16, 720.29], [597.38, 649.92], [1027.91, 695.64], [951.49, 759.05], [829.11, 982.56], [617.07, 964.13]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[800.57, 771.22], [684.42, 599.53], [1031.21, 681.53], [948.01, 728.08], [730.38, 973.04], [679.46, 1022.81]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[748.99, 822.14], [771.46, 549.14], [924.14, 760.44], [907.27, 717.76], [631.65, 963.52], [766.35, 1006.11]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[697.53, 872.94], [858.28, 498.88], [833.59, 848.55], [858.52, 702.76], [533.17, 954.02], [853.02, 989.45]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[645.94, 923.87], [867.57, 504.97], [742.82, 936.88], [809.64, 687.72], [491.2, 944.5], [939.91, 972.75]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[594.36, 974.8], [792.74, 572.18], [657.12, 1009.61], [760.76, 672.68], [589.94, 934.98], [933.18, 888.04]]) <= 50.0"
] | ballsim | G_3_864 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
162,
99,
228
],
"density": 1,
"position": [
503,
753
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-973.71,
-3.25
]
},
{
"color": [
235,
236,
79
],
"density": 1,
"position": [
702,
817
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
260.06,
-830.26
]
},
{
"color": [
219,
85,
125
],
"density": 1,
"position": [
755,
704
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
31.18,
850.81
]
},
{
"color": [
64,
100,
90
],
"density": 1,
"position": [
687,
751
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
685.36,
622.09
]
},
{
"color": [
211,
150,
148
],
"density": 1,
"position": [
822,
607
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-688.3,
489.24
]
},
{
"color": [
163,
183,
52
],
"density": 1,
"position": [
870,
528
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
109.67,
-877.92
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 864,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 9201403
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (716, 844), initial velocity (863.16, -486.47) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (742, 713), initial velocity (900.4, 429.93) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (933, 705), initial velocity (489.73, 703.83) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (732, 785), initial velocity (364.23, 759.94) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (832, 808), initial velocity (-409.63, 859.15) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[785.58, 928.47], [832.04, 755.99], [981.97, 775.39], [768.43, 771.41], [807.78, 877.16]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[773.81, 1043.58], [922.08, 798.99], [1004.01, 818.84], [804.85, 738.96], [827.31, 902.58]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[662.7, 975.7], [863.5, 842.09], [1014.77, 769.74], [841.37, 706.43], [846.89, 928.06]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[636.87, 822.99], [746.04, 885.08], [1057.74, 720.76], [877.79, 673.99], [866.42, 953.47]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[611.04, 670.28], [628.58, 928.07], [1021.57, 750.92], [914.22, 641.54], [876.1, 969.03]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[627.44, 559.81], [623.4, 858.79], [972.6, 793.89], [950.65, 609.09], [850.68, 949.5]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[780.15, 585.64], [666.39, 741.34], [923.62, 836.86], [924.31, 639.41], [825.27, 929.97]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[932.86, 611.47], [709.39, 623.88], [874.65, 879.83], [891.87, 675.83], [799.85, 910.44]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[992.28, 730.59], [752.38, 506.43], [825.68, 922.79], [859.42, 712.26], [774.44, 890.9]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[971.69, 836.88], [743.96, 440.39], [776.7, 965.76], [826.97, 748.68], [749.02, 871.37]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[818.98, 811.05], [668.95, 525.82], [727.73, 1008.73], [794.53, 785.11], [723.61, 851.84]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[767.67, 886.61], [625.96, 643.27], [704.65, 1025.8], [684.04, 743.47], [698.19, 832.31]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[713.65, 949.7], [525.01, 659.86], [751.35, 988.74], [601.53, 750.9], [672.84, 812.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[633.36, 932.49], [575.16, 769.18], [820.17, 1019.28], [547.48, 816.35], [647.43, 793.3]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[560.29, 908.04], [600.53, 735.14], [799.96, 960.79], [593.41, 841.09], [659.24, 811.01]]) <= 50.0"
] | ballsim | G_3_760 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
132,
212,
172
],
"density": 1,
"position": [
716,
844
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
863.16,
-486.47
]
},
{
"color": [
220,
105,
169
],
"density": 1,
"position": [
742,
713
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
900.4,
429.93
]
},
{
"color": [
222,
227,
225
],
"density": 1,
"position": [
933,
705
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
489.73,
703.83
]
},
{
"color": [
71,
60,
167
],
"density": 1,
"position": [
732,
785
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
364.23,
759.94
]
},
{
"color": [
51,
180,
59
],
"density": 1,
"position": [
832,
808
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-409.63,
859.15
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 760,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7035245
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (835, 924), initial velocity (817.92, -165.29) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (777, 684), initial velocity (209.34, 921.16) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (837, 744), initial velocity (11.43, 834.71) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (790, 833), initial velocity (-588.77, -747.19) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (954, 952), initial velocity (-658.2, -736.5) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (602, 757), initial velocity (602.82, -728.4) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[875.37, 907.47], [797.92, 776.11], [838.15, 827.47], [731.11, 758.29], [964.0, 878.34], [662.28, 684.15]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[877.71, 915.79], [806.44, 846.74], [817.81, 895.36], [672.23, 683.59], [1065.56, 791.1], [722.56, 611.3]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[911.78, 955.86], [761.11, 824.19], [794.71, 986.34], [613.19, 608.69], [1033.66, 664.7], [782.99, 538.26]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[945.76, 995.82], [715.9, 801.71], [771.66, 1032.43], [554.31, 533.99], [907.31, 593.19], [843.26, 465.42]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[920.11, 1001.36], [670.68, 779.22], [748.62, 941.68], [601.03, 520.24], [780.97, 521.68], [903.54, 495.96]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[868.49, 991.91], [625.47, 756.74], [725.57, 850.94], [695.17, 533.86], [654.63, 450.17], [963.82, 568.81]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[816.88, 982.46], [580.25, 734.25], [702.52, 760.19], [789.31, 547.49], [539.27, 512.82], [1024.1, 641.66]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[765.27, 973.01], [535.04, 711.77], [679.48, 669.44], [883.45, 561.11], [540.52, 657.98], [1084.4, 714.51]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[713.66, 963.57], [438.49, 778.18], [656.43, 578.69], [977.59, 574.73], [593.07, 714.25], [1052.79, 765.71]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[662.05, 954.12], [522.77, 751.35], [633.38, 487.95], [981.62, 640.39], [655.28, 753.81], [959.58, 781.51]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[610.44, 944.67], [647.87, 700.96], [610.33, 493.28], [946.35, 728.74], [717.49, 793.36], [866.37, 797.3]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[558.83, 935.22], [772.97, 650.57], [587.29, 584.03], [911.07, 817.1], [770.39, 849.03], [778.24, 804.28]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[507.35, 925.8], [897.75, 600.31], [564.3, 674.55], [875.88, 905.23], [795.59, 952.31], [705.37, 785.14]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[513.16, 883.2], [972.78, 578.82], [541.25, 765.3], [840.6, 993.59], [820.86, 1055.86], [632.32, 765.95]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[545.61, 879.4], [866.58, 661.95], [500.59, 801.81], [861.22, 985.1], [743.81, 982.18], [559.28, 746.76]]) <= 50.0"
] | ballsim | G_3_194 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
153,
72,
59
],
"density": 1,
"position": [
835,
924
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
817.92,
-165.29
]
},
{
"color": [
78,
178,
109
],
"density": 1,
"position": [
777,
684
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
209.34,
921.16
]
},
{
"color": [
236,
148,
212
],
"density": 1,
"position": [
837,
744
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
11.43,
834.71
]
},
{
"color": [
147,
144,
166
],
"density": 1,
"position": [
790,
833
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-588.77,
-747.19
]
},
{
"color": [
207,
194,
223
],
"density": 1,
"position": [
954,
952
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-658.2,
-736.5
]
},
{
"color": [
248,
145,
140
],
"density": 1,
"position": [
602,
757
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
602.82,
-728.4
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 194,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 489425
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (705, 718), initial velocity (-369.49, 909.38) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (829, 759), initial velocity (325.01, 850.29) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (748, 789), initial velocity (778.11, -564.04) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (681, 630), initial velocity (-637.76, -534.57) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (685, 884), initial velocity (12.41, 900.66) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (653, 735), initial velocity (-763.41, 316.72) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[668.06, 808.94], [861.5, 844.03], [825.81, 732.6], [617.23, 576.53], [686.25, 974.06], [580.72, 766.67]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[631.12, 899.88], [834.41, 825.87], [903.62, 676.21], [603.92, 523.07], [614.64, 937.92], [657.07, 798.33]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[687.04, 955.53], [776.86, 755.04], [930.43, 708.33], [633.91, 528.26], [622.96, 911.1], [733.6, 830.07]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[736.57, 867.92], [719.47, 684.39], [920.49, 803.91], [619.51, 610.22], [662.95, 884.85], [809.94, 861.74]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[755.13, 726.66], [662.07, 613.73], [895.87, 843.63], [605.1, 692.18], [702.94, 858.61], [852.28, 851.13]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[773.68, 585.4], [604.67, 543.08], [935.52, 804.45], [590.7, 774.13], [742.93, 832.36], [793.41, 769.17]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[683.73, 632.08], [610.97, 484.15], [975.17, 765.26], [581.09, 856.09], [782.92, 806.12], [734.55, 687.21]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[574.36, 718.78], [586.55, 569.19], [1014.82, 726.08], [595.49, 938.05], [822.91, 779.87], [675.69, 605.25]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[687.42, 805.47], [619.04, 654.22], [1025.07, 737.79], [609.89, 1020.01], [862.9, 753.63], [616.83, 523.29]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[800.48, 892.17], [651.54, 739.26], [1010.96, 791.7], [607.11, 973.72], [902.89, 727.38], [582.05, 483.06]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[811.06, 801.36], [684.03, 824.29], [968.1, 795.83], [670.88, 920.26], [942.88, 701.14], [623.6, 575.0]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[792.5, 660.1], [716.53, 833.94], [914.34, 781.09], [734.65, 949.17], [969.32, 698.37], [665.16, 666.94]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[739.6, 578.76], [748.94, 786.75], [860.72, 766.38], [690.82, 854.02], [966.59, 746.0], [706.61, 758.66]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[607.99, 633.33], [778.44, 739.43], [812.94, 751.63], [643.75, 753.21], [963.86, 793.76], [748.16, 850.6]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[634.45, 582.64], [753.44, 692.12], [874.2, 736.89], [615.37, 709.9], [944.24, 812.27], [771.03, 910.2]]) <= 50.0"
] | ballsim | G_3_890 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
124,
250,
205
],
"density": 1,
"position": [
705,
718
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-369.49,
909.38
]
},
{
"color": [
231,
236,
59
],
"density": 1,
"position": [
829,
759
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
325.01,
850.29
]
},
{
"color": [
151,
229,
86
],
"density": 1,
"position": [
748,
789
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
778.11,
-564.04
]
},
{
"color": [
216,
120,
93
],
"density": 1,
"position": [
681,
630
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-637.76,
-534.57
]
},
{
"color": [
89,
68,
160
],
"density": 1,
"position": [
685,
884
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
12.41,
900.66
]
},
{
"color": [
91,
244,
152
],
"density": 1,
"position": [
653,
735
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-763.41,
316.72
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 890,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3334775
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (685, 512), initial velocity (832.45, 325.63) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (540, 716), initial velocity (-820.27, 153.21) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (920, 947), initial velocity (16.0, 807.28) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (620, 702), initial velocity (529.14, 779.57) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (907, 677), initial velocity (606.91, 713.55) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[768.25, 544.57], [457.97, 731.33], [921.61, 1027.74], [672.91, 779.95], [967.69, 748.36]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[851.5, 577.14], [411.89, 767.41], [923.22, 1003.0], [725.81, 857.91], [1028.39, 819.72]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[934.96, 609.79], [460.54, 834.28], [924.84, 922.06], [778.85, 936.06], [990.53, 834.27]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1018.22, 642.35], [514.82, 897.66], [926.45, 841.32], [831.75, 1014.01], [898.4, 817.38]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[1041.73, 709.41], [569.09, 961.04], [928.06, 760.59], [884.66, 1022.87], [806.26, 800.48]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[1028.3, 797.79], [623.37, 1024.42], [929.67, 679.85], [937.56, 944.92], [714.12, 783.59]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[1025.97, 885.46], [677.64, 1023.66], [931.28, 599.11], [979.21, 866.96], [621.98, 766.69]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[921.5, 881.29], [731.91, 960.28], [932.89, 518.38], [968.73, 789.01], [529.84, 749.8]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[817.03, 877.12], [786.18, 896.9], [919.73, 446.18], [958.26, 711.06], [437.7, 732.9]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[844.3, 777.23], [708.75, 929.24], [849.03, 481.34], [947.79, 633.1], [449.17, 775.84]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[871.89, 677.1], [630.99, 961.81], [778.32, 520.32], [937.31, 555.15], [509.86, 847.2]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[899.47, 576.98], [553.23, 994.38], [707.62, 559.31], [926.84, 477.19], [570.56, 918.56]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[926.99, 524.05], [579.43, 969.84], [637.09, 598.2], [916.39, 442.75], [654.24, 972.93]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[954.58, 609.89], [589.49, 960.2], [566.39, 637.19], [905.92, 534.63], [782.71, 995.05]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[982.17, 695.73], [599.55, 950.56], [495.68, 676.18], [895.44, 626.86], [911.18, 1017.17]]) <= 50.0"
] | ballsim | G_3_573 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
226,
244,
141
],
"density": 1,
"position": [
685,
512
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
832.45,
325.63
]
},
{
"color": [
235,
226,
118
],
"density": 1,
"position": [
540,
716
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-820.27,
153.21
]
},
{
"color": [
72,
50,
57
],
"density": 1,
"position": [
920,
947
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
16,
807.28
]
},
{
"color": [
117,
208,
204
],
"density": 1,
"position": [
620,
702
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
529.14,
779.57
]
},
{
"color": [
123,
141,
111
],
"density": 1,
"position": [
907,
677
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
606.91,
713.55
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 573,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4940944
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (850, 724), initial velocity (911.54, -379.14) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (807, 774), initial velocity (-113.7, -937.68) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (831, 834), initial velocity (590.28, -565.42) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (708, 950), initial velocity (618.59, 532.4) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (889, 864), initial velocity (33.58, 806.17) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[941.16, 686.08], [795.62, 680.23], [890.03, 777.46], [769.87, 1003.25], [892.37, 944.62]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[1003.16, 677.34], [784.25, 586.45], [949.07, 720.91], [793.3, 1018.05], [824.55, 954.05]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[1053.53, 731.73], [772.84, 492.44], [914.65, 718.25], [730.68, 984.48], [752.98, 922.23]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1006.73, 799.46], [753.67, 406.48], [854.79, 730.34], [657.63, 983.58], [692.16, 857.93]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[890.21, 797.45], [757.23, 492.43], [794.93, 742.42], [635.63, 931.63], [631.35, 793.62]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[773.68, 795.45], [768.61, 586.2], [735.06, 754.51], [634.72, 858.59], [570.53, 729.31]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[678.93, 815.23], [779.99, 679.98], [658.45, 749.84], [633.82, 785.54], [509.72, 665.01]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[602.39, 853.21], [791.36, 773.75], [643.67, 676.05], [550.03, 772.72], [552.25, 704.05]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[536.22, 880.84], [802.74, 867.53], [633.88, 598.63], [460.8, 763.86], [616.56, 764.87]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[574.21, 804.3], [814.12, 961.3], [631.12, 528.24], [434.33, 745.01], [680.87, 825.68]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[612.19, 727.76], [799.91, 1029.49], [708.54, 538.03], [523.57, 753.87], [745.17, 886.5]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[650.18, 651.22], [706.14, 1018.12], [785.96, 547.82], [612.8, 762.73], [809.48, 947.31]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[688.08, 574.88], [659.1, 960.27], [863.18, 557.59], [701.81, 771.57], [841.81, 976.16]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[726.06, 498.34], [647.72, 866.49], [915.04, 592.93], [791.04, 780.44], [780.99, 911.85]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[764.05, 421.78], [636.35, 772.72], [924.83, 670.35], [880.28, 789.3], [720.18, 847.55]]) <= 50.0"
] | ballsim | G_3_517 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
100,
53,
176
],
"density": 1,
"position": [
850,
724
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
911.54,
-379.14
]
},
{
"color": [
171,
208,
52
],
"density": 1,
"position": [
807,
774
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-113.7,
-937.68
]
},
{
"color": [
95,
254,
59
],
"density": 1,
"position": [
831,
834
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
590.28,
-565.42
]
},
{
"color": [
81,
99,
226
],
"density": 1,
"position": [
708,
950
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
618.59,
532.4
]
},
{
"color": [
209,
255,
144
],
"density": 1,
"position": [
889,
864
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
33.58,
806.17
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 517,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6292005
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (888, 562), initial velocity (-198.57, 928.04) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (966, 595), initial velocity (886.33, -428.7) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (666, 937), initial velocity (-779.75, 186.92) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (832, 724), initial velocity (769.23, -476.03) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (750, 506), initial velocity (980.78, -2.55) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[868.15, 654.8], [958.81, 607.45], [588.02, 955.7], [908.93, 676.39], [848.07, 505.74]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[848.3, 747.6], [877.36, 662.77], [536.28, 959.25], [985.86, 628.79], [946.14, 505.49]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[828.4, 840.63], [795.71, 718.23], [591.6, 900.93], [979.69, 629.15], [923.27, 575.2]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[808.56, 933.42], [714.27, 773.56], [646.77, 842.75], [905.3, 677.27], [869.93, 655.94]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[788.71, 1026.22], [754.66, 740.37], [635.39, 832.92], [866.53, 761.01], [789.16, 709.26]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[768.86, 995.82], [729.91, 883.97], [589.25, 848.35], [827.76, 844.75], [772.79, 651.04]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[749.01, 903.02], [695.71, 1043.96], [543.11, 863.78], [788.99, 928.49], [761.14, 584.63]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[729.16, 810.22], [661.5, 910.9], [496.96, 879.21], [750.22, 1012.23], [749.5, 518.23]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[709.31, 717.43], [627.3, 750.92], [496.83, 868.07], [711.46, 1013.53], [737.85, 451.83]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[689.46, 624.63], [593.1, 590.93], [533.27, 835.82], [672.69, 929.78], [726.21, 499.7]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[669.62, 531.83], [589.38, 448.55], [569.72, 803.57], [633.92, 846.04], [714.56, 566.11]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[703.58, 532.24], [637.42, 486.96], [571.73, 746.3], [636.09, 792.05], [702.92, 632.51]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[795.58, 633.32], [569.09, 537.2], [540.9, 665.33], [677.3, 766.55], [691.3, 698.75]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[887.82, 734.67], [520.74, 569.31], [516.52, 604.25], [718.61, 810.49], [679.65, 711.66]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[980.06, 836.01], [616.16, 503.95], [531.58, 664.56], [759.91, 888.86], [668.01, 698.06]]) <= 50.0"
] | ballsim | G_3_5 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
164,
210,
68
],
"density": 1,
"position": [
888,
562
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-198.57,
928.04
]
},
{
"color": [
100,
121,
172
],
"density": 1,
"position": [
966,
595
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
886.33,
-428.7
]
},
{
"color": [
124,
162,
60
],
"density": 1,
"position": [
666,
937
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-779.75,
186.92
]
},
{
"color": [
218,
111,
123
],
"density": 1,
"position": [
832,
724
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
769.23,
-476.03
]
},
{
"color": [
231,
154,
123
],
"density": 1,
"position": [
750,
506
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
980.78,
-2.55
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 5,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4449895
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (815, 770), initial velocity (995.14, 31.31) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (708, 842), initial velocity (-589.32, 605.95) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (775, 806), initial velocity (825.61, -244.82) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (811, 905), initial velocity (-43.07, -972.28) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (909, 638), initial velocity (594.65, 547.77) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[914.51, 773.12], [649.06, 902.6], [857.57, 781.51], [806.7, 807.78], [968.47, 692.79]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[1008.67, 780.14], [590.13, 963.19], [940.14, 757.03], [802.41, 710.57], [1032.85, 744.01]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[1051.97, 819.5], [556.39, 945.95], [1022.91, 732.48], [798.1, 613.11], [1077.51, 717.37]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[1005.43, 811.1], [544.33, 862.28], [972.2, 760.02], [793.8, 515.89], [1010.86, 650.26]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[979.77, 866.93], [532.27, 778.62], [866.19, 749.67], [800.98, 453.97], [944.84, 588.08]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[954.11, 922.77], [520.21, 694.95], [760.18, 739.32], [836.69, 524.8], [892.64, 533.87]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[928.46, 978.6], [508.14, 611.28], [654.18, 728.96], [718.77, 506.93], [954.66, 550.9]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[896.94, 1030.18], [497.8, 532.89], [548.17, 718.61], [606.98, 507.91], [947.65, 618.09]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[851.78, 1071.83], [511.18, 626.59], [491.09, 708.26], [553.04, 569.2], [940.64, 685.28]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[831.95, 1035.48], [472.34, 758.23], [597.1, 697.91], [561.17, 585.39], [933.64, 752.46]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[819.89, 975.22], [492.12, 889.87], [703.1, 687.56], [569.3, 601.58], [926.63, 819.65]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[807.83, 914.97], [554.89, 947.87], [809.11, 677.21], [577.43, 617.76], [919.62, 886.84]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[795.8, 854.86], [663.41, 864.41], [914.85, 666.88], [585.54, 633.91], [912.63, 953.86]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[808.39, 794.61], [747.55, 780.74], [1020.86, 656.53], [593.67, 650.1], [905.43, 1020.9]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[917.17, 734.36], [735.49, 697.08], [996.1, 741.19], [601.8, 666.28], [843.69, 1048.34]]) <= 50.0"
] | ballsim | G_3_201 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
131,
211,
190
],
"density": 1,
"position": [
815,
770
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
995.14,
31.31
]
},
{
"color": [
189,
151,
146
],
"density": 1,
"position": [
708,
842
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-589.32,
605.95
]
},
{
"color": [
65,
73,
85
],
"density": 1,
"position": [
775,
806
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
825.61,
-244.82
]
},
{
"color": [
119,
182,
185
],
"density": 1,
"position": [
811,
905
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-43.07,
-972.28
]
},
{
"color": [
177,
93,
229
],
"density": 1,
"position": [
909,
638
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
594.65,
547.77
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 201,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7849990
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (754, 770), initial velocity (71.51, -800.37) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (738, 879), initial velocity (402.11, -889.26) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (536, 854), initial velocity (741.22, -389.33) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (844, 819), initial velocity (74.62, -819.71) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (969, 688), initial velocity (847.74, 146.66) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (862, 609), initial velocity (-109.92, 797.05) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[761.15, 689.97], [778.21, 790.08], [610.12, 815.06], [851.47, 737.02], [1053.77, 702.67], [851.01, 688.71]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[760.62, 609.94], [818.42, 701.17], [684.24, 776.12], [887.66, 743.45], [1059.93, 762.72], [801.51, 606.6]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[713.02, 529.71], [858.73, 612.03], [758.55, 737.08], [927.56, 761.02], [1030.17, 843.65], [831.41, 503.93]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[665.54, 449.67], [898.94, 523.11], [832.67, 698.14], [967.35, 778.55], [1000.48, 924.39], [861.25, 483.63]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[618.05, 518.91], [914.41, 448.47], [906.79, 659.2], [1007.15, 796.08], [951.14, 993.79], [891.08, 586.05]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[570.56, 598.94], [817.29, 446.31], [980.91, 620.26], [1046.94, 813.61], [896.06, 1051.59], [920.92, 688.46]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[523.08, 678.97], [720.17, 455.96], [983.89, 622.4], [1043.86, 806.39], [840.98, 985.51], [950.75, 790.88]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[475.59, 759.0], [623.05, 465.6], [913.11, 667.13], [1008.79, 780.7], [785.9, 919.42], [980.58, 893.3]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[460.01, 820.61], [567.5, 499.24], [842.34, 711.85], [973.7, 755.02], [730.82, 853.34], [939.48, 954.76]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[553.07, 819.51], [607.71, 588.16], [771.56, 756.58], [938.62, 729.34], [675.75, 787.25], [835.87, 980.12]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[646.14, 818.41], [647.92, 677.08], [700.78, 801.31], [903.54, 703.65], [620.67, 721.16], [732.25, 1005.49]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[640.84, 817.31], [688.13, 765.99], [810.04, 846.03], [868.45, 677.97], [565.59, 655.07], [628.64, 1030.86]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[618.19, 816.22], [728.24, 854.69], [950.8, 890.65], [833.46, 652.35], [510.65, 589.15], [595.04, 1015.91]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[595.49, 815.12], [768.45, 943.6], [950.98, 854.01], [798.38, 626.67], [565.72, 586.67], [668.81, 938.86]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[572.78, 814.02], [808.66, 1032.52], [841.68, 754.15], [763.29, 600.98], [650.48, 601.34], [742.59, 861.81]]) <= 50.0"
] | ballsim | G_3_120 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
168,
186,
180
],
"density": 1,
"position": [
754,
770
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
71.51,
-800.37
]
},
{
"color": [
188,
156,
96
],
"density": 1,
"position": [
738,
879
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
402.11,
-889.26
]
},
{
"color": [
70,
201,
96
],
"density": 1,
"position": [
536,
854
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
741.22,
-389.33
]
},
{
"color": [
190,
139,
227
],
"density": 1,
"position": [
844,
819
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
74.62,
-819.71
]
},
{
"color": [
120,
238,
72
],
"density": 1,
"position": [
969,
688
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
847.74,
146.66
]
},
{
"color": [
105,
61,
144
],
"density": 1,
"position": [
862,
609
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-109.92,
797.05
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 120,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2068750
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (460, 747), initial velocity (847.36, 214.45) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (814, 770), initial velocity (813.72, 283.05) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (874, 490), initial velocity (624.42, 513.85) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (716, 560), initial velocity (-399.33, 821.24) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (665, 732), initial velocity (699.84, -583.48) m/s\n- Ball 6: regular polygon (5 sides), radius 20.0m, initial position (753, 812), initial velocity (-702.79, 407.32) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[544.74, 768.44], [895.37, 798.3], [936.45, 541.39], [676.06, 642.13], [735.0, 673.65], [682.71, 852.72]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[629.48, 789.87], [976.74, 826.59], [998.9, 592.78], [636.12, 724.26], [804.99, 615.3], [612.42, 893.45]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[714.44, 811.36], [1037.43, 842.9], [1035.19, 659.48], [596.08, 806.59], [875.16, 556.81], [541.96, 934.27]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[799.18, 832.8], [972.22, 786.58], [1048.47, 739.25], [556.13, 888.72], [945.16, 498.46], [555.98, 926.32]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[883.92, 854.23], [907.01, 730.25], [1061.75, 819.01], [495.4, 912.94], [881.54, 517.24], [642.95, 914.47]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[968.66, 875.67], [841.8, 673.93], [991.11, 850.34], [553.5, 862.28], [796.02, 548.69], [731.4, 905.19]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[976.79, 967.44], [776.59, 617.61], [1009.73, 784.38], [611.6, 811.62], [710.49, 580.14], [819.85, 895.92]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[889.19, 1004.23], [711.38, 561.28], [1028.6, 718.17], [669.71, 760.96], [624.97, 611.58], [908.3, 886.64]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[801.59, 1041.03], [646.17, 504.96], [1047.45, 651.96], [727.82, 710.3], [539.45, 643.03], [996.76, 877.36]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[713.99, 1037.01], [580.96, 448.64], [982.09, 634.38], [785.92, 659.64], [453.93, 674.47], [1012.62, 826.18]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[626.4, 1000.21], [557.37, 516.83], [915.32, 617.6], [844.03, 608.98], [450.95, 753.56], [976.44, 744.96]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[538.8, 963.42], [541.2, 601.45], [855.37, 612.63], [891.63, 540.15], [466.49, 843.33], [940.26, 663.73]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[555.32, 912.99], [525.08, 685.86], [801.77, 618.4], [929.58, 454.97], [476.99, 885.22], [904.17, 582.71]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[672.92, 895.51], [508.92, 770.48], [748.03, 624.19], [844.22, 444.31], [477.61, 833.05], [867.99, 501.48]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[790.52, 878.02], [544.04, 817.84], [694.29, 629.98], [751.26, 454.05], [447.89, 806.76], [831.81, 468.28]]) <= 50.0"
] | ballsim | G_3_709 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
246,
131,
108
],
"density": 1,
"position": [
460,
747
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
847.36,
214.45
]
},
{
"color": [
180,
250,
176
],
"density": 1,
"position": [
814,
770
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
813.72,
283.05
]
},
{
"color": [
199,
186,
128
],
"density": 1,
"position": [
874,
490
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
624.42,
513.85
]
},
{
"color": [
168,
136,
102
],
"density": 1,
"position": [
716,
560
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-399.33,
821.24
]
},
{
"color": [
96,
249,
126
],
"density": 1,
"position": [
665,
732
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
699.84,
-583.48
]
},
{
"color": [
114,
160,
236
],
"density": 1,
"position": [
753,
812
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-702.79,
407.32
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 709,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6735116
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (620, 893), initial velocity (-172.9, 787.51) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (935, 953), initial velocity (-892.7, 171.07) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (750, 800), initial velocity (753.82, 423.41) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (941, 691), initial velocity (-559.4, -590.55) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (800, 753), initial velocity (225.54, -780.82) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (673, 554), initial velocity (152.6, -932.12) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[602.71, 971.76], [845.74, 970.11], [825.39, 842.33], [885.07, 631.94], [822.56, 674.92], [694.31, 479.4]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[616.0, 956.41], [756.48, 987.23], [900.78, 884.67], [847.4, 554.62], [823.4, 618.56], [761.45, 545.84]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[648.38, 882.35], [667.0, 1004.39], [976.3, 927.06], [836.0, 450.74], [840.36, 640.84], [781.29, 564.98]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[680.68, 808.47], [592.73, 975.37], [912.72, 868.44], [869.67, 485.8], [904.5, 710.27], [753.87, 536.88]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[712.98, 734.6], [530.57, 909.07], [849.15, 809.82], [921.35, 576.31], [968.64, 779.71], [726.45, 508.78]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[745.28, 660.72], [468.4, 842.76], [785.57, 751.21], [973.03, 666.81], [1032.77, 849.14], [699.04, 480.67]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[777.58, 586.84], [519.38, 776.45], [722.0, 692.59], [1024.72, 757.31], [947.15, 809.76], [685.68, 495.84]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[809.88, 512.97], [581.54, 710.14], [658.43, 633.97], [1020.79, 807.39], [861.29, 770.21], [680.02, 534.7]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[842.18, 439.09], [643.7, 643.83], [594.85, 575.35], [918.74, 786.2], [775.43, 730.66], [674.35, 573.57]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[875.46, 459.24], [652.27, 631.11], [534.42, 526.41], [816.68, 765.01], [689.56, 691.11], [732.4, 548.72]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[892.75, 537.99], [603.8, 612.22], [517.45, 611.18], [714.63, 743.82], [641.28, 689.14], [820.71, 493.62]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[910.03, 616.75], [531.53, 569.54], [500.48, 695.94], [635.18, 745.24], [594.41, 688.59], [869.31, 467.36]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[927.28, 695.31], [468.53, 554.94], [483.54, 780.5], [592.42, 783.16], [504.28, 644.66], [789.82, 534.15]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[944.56, 774.07], [511.34, 567.69], [466.7, 865.26], [549.55, 821.17], [494.53, 677.0], [710.13, 601.09]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[961.85, 852.83], [582.19, 529.33], [483.68, 950.03], [506.68, 859.18], [540.33, 770.09], [630.45, 668.03]]) <= 50.0"
] | ballsim | G_3_504 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
146,
167,
116
],
"density": 1,
"position": [
620,
893
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-172.9,
787.51
]
},
{
"color": [
51,
148,
65
],
"density": 1,
"position": [
935,
953
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-892.7,
171.07
]
},
{
"color": [
59,
88,
94
],
"density": 1,
"position": [
750,
800
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
753.82,
423.41
]
},
{
"color": [
53,
121,
189
],
"density": 1,
"position": [
941,
691
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-559.4,
-590.55
]
},
{
"color": [
222,
231,
64
],
"density": 1,
"position": [
800,
753
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
225.54,
-780.82
]
},
{
"color": [
151,
142,
111
],
"density": 1,
"position": [
673,
554
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
152.6,
-932.12
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 504,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2799249
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (841, 871), initial velocity (-265.33, 839.64) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (757, 710), initial velocity (-883.29, 54.96) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (757, 849), initial velocity (480.15, 750.96) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (705, 662), initial velocity (-446.05, 824.97) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (692, 778), initial velocity (966.99, -154.14) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[844.25, 882.56], [706.22, 742.77], [739.5, 924.1], [628.82, 721.55], [788.7, 762.59]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[893.68, 855.72], [688.77, 799.76], [656.4, 993.92], [632.76, 760.73], [885.41, 747.19]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[937.22, 818.39], [671.27, 856.88], [610.64, 1020.06], [737.22, 800.01], [982.35, 731.75]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[980.65, 781.16], [670.68, 897.0], [622.67, 926.18], [841.42, 839.2], [1055.55, 757.06]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[1014.57, 753.44], [769.8, 837.4], [607.07, 909.07], [904.55, 807.25], [1000.78, 716.36]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[1027.71, 746.51], [866.23, 786.07], [671.79, 891.95], [859.1, 738.32], [969.21, 693.25]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[1040.84, 739.57], [924.73, 843.76], [736.51, 874.84], [771.09, 628.46], [933.61, 764.53]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[1053.98, 732.64], [827.44, 836.46], [801.23, 857.72], [683.08, 518.59], [961.57, 780.62]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[1055.47, 745.92], [830.29, 729.02], [766.41, 929.37], [612.46, 495.6], [1013.57, 772.66]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[1071.31, 752.21], [833.22, 621.5], [682.91, 916.89], [679.98, 598.37], [1025.51, 774.75]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[1042.88, 741.13], [758.84, 647.88], [599.42, 904.42], [731.13, 729.53], [1003.34, 748.46]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[1016.64, 720.8], [667.19, 704.18], [649.1, 891.94], [782.28, 860.68], [972.2, 722.17]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[993.53, 700.51], [589.25, 760.34], [727.55, 882.29], [755.7, 844.93], [937.52, 695.94]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[967.64, 684.89], [680.9, 816.63], [728.41, 917.52], [757.97, 682.94], [902.75, 669.64]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[938.44, 674.98], [772.55, 872.93], [729.26, 952.75], [724.69, 582.53], [867.99, 643.35]]) <= 50.0"
] | ballsim | G_3_738 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
58,
96,
227
],
"density": 1,
"position": [
841,
871
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-265.33,
839.64
]
},
{
"color": [
237,
201,
107
],
"density": 1,
"position": [
757,
710
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-883.29,
54.96
]
},
{
"color": [
253,
209,
123
],
"density": 1,
"position": [
757,
849
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
480.15,
750.96
]
},
{
"color": [
63,
144,
69
],
"density": 1,
"position": [
705,
662
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-446.05,
824.97
]
},
{
"color": [
159,
175,
248
],
"density": 1,
"position": [
692,
778
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
966.99,
-154.14
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 738,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2638753
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (716, 721), initial velocity (-874.76, 70.49) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (687, 765), initial velocity (16.16, -802.19) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (651, 1003), initial velocity (170.94, 943.76) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (891, 822), initial velocity (599.67, 578.3) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (766, 937), initial velocity (-838.5, -289.3) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[712.42, 667.09], [596.94, 751.38], [712.19, 961.69], [950.96, 879.84], [682.16, 908.07]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[720.33, 604.84], [494.33, 746.89], [781.5, 895.38], [955.57, 897.45], [598.32, 879.14]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[728.26, 542.43], [534.16, 742.39], [850.99, 828.9], [881.85, 858.2], [514.28, 850.14]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[736.17, 480.17], [636.77, 737.9], [872.51, 714.83], [856.1, 866.8], [482.83, 821.21]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[765.0, 482.31], [739.38, 733.4], [883.96, 590.68], [840.43, 885.48], [566.68, 792.27]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[807.99, 528.02], [841.99, 728.91], [895.41, 466.53], [824.75, 904.16], [650.51, 763.34]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[850.99, 573.72], [944.6, 724.42], [782.15, 446.69], [809.08, 922.83], [734.35, 734.41]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[893.98, 619.42], [1047.22, 719.93], [699.9, 540.39], [793.41, 941.51], [818.19, 705.48]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[936.97, 665.13], [1058.25, 781.99], [617.65, 634.09], [777.73, 960.19], [902.03, 676.55]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[1021.17, 687.05], [1008.62, 868.28], [535.4, 727.79], [762.06, 978.86], [903.46, 695.2]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[1061.82, 740.74], [928.24, 932.22], [480.1, 821.5], [746.38, 997.54], [903.36, 714.73]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[1042.38, 818.28], [847.87, 996.16], [562.35, 915.2], [730.71, 1016.22], [903.27, 734.26]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[961.22, 850.77], [769.01, 1055.92], [644.39, 1008.66], [715.08, 1034.86], [903.17, 753.74]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[879.85, 883.34], [763.01, 956.94], [703.36, 958.43], [746.37, 1046.11], [903.07, 773.27]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[816.34, 926.22], [797.32, 846.69], [668.25, 870.85], [814.63, 1051.48], [902.97, 792.81]]) <= 50.0"
] | ballsim | G_3_262 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
104,
69,
148
],
"density": 1,
"position": [
716,
721
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-874.76,
70.49
]
},
{
"color": [
72,
157,
138
],
"density": 1,
"position": [
687,
765
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
16.16,
-802.19
]
},
{
"color": [
230,
116,
127
],
"density": 1,
"position": [
651,
1003
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
170.94,
943.76
]
},
{
"color": [
244,
244,
206
],
"density": 1,
"position": [
891,
822
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
599.67,
578.3
]
},
{
"color": [
219,
150,
211
],
"density": 1,
"position": [
766,
937
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-838.5,
-289.3
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 262,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5687406
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (705, 710), initial velocity (-884.02, -194.06) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (686, 965), initial velocity (668.54, 714.37) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (789, 836), initial velocity (761.65, 449.68) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (773, 699), initial velocity (-273.08, -930.76) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (667, 593), initial velocity (211.87, -804.12) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (594, 569), initial velocity (925.42, -360.52) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[616.6, 690.59], [752.85, 1036.44], [865.17, 880.97], [745.68, 605.93], [720.89, 531.46], [644.06, 508.42]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[528.19, 671.18], [747.11, 1035.29], [929.04, 913.64], [722.47, 525.52], [805.34, 487.57], [715.2, 475.55]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[500.61, 712.77], [675.5, 968.28], [883.96, 837.28], [727.05, 530.59], [791.23, 542.33], [739.08, 441.8]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[520.02, 801.17], [604.07, 901.44], [838.99, 761.11], [747.15, 583.41], [747.34, 626.78], [675.03, 512.02]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[541.91, 862.55], [581.7, 834.59], [806.05, 696.97], [769.98, 665.67], [703.44, 711.23], [609.8, 549.31]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[630.31, 821.39], [631.39, 767.74], [894.95, 754.66], [690.34, 645.46], [659.54, 795.68], [586.61, 628.63]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[568.15, 867.16], [681.07, 700.9], [983.84, 812.35], [610.71, 625.24], [766.21, 793.21], [563.42, 707.95]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[567.03, 850.61], [730.75, 634.05], [984.94, 782.26], [547.92, 621.88], [875.93, 788.99], [540.22, 787.27]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[614.57, 785.4], [780.44, 567.21], [927.25, 693.37], [568.14, 701.52], [985.64, 784.77], [517.03, 866.59]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[662.1, 720.19], [830.12, 500.36], [869.56, 604.47], [588.35, 781.16], [1067.13, 752.31], [590.93, 848.82]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[709.63, 654.98], [772.8, 509.12], [822.06, 525.77], [608.57, 860.8], [990.25, 723.69], [670.25, 825.63]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[757.17, 589.77], [671.98, 524.84], [815.33, 487.84], [628.78, 940.44], [880.54, 727.92], [749.57, 802.44]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[804.58, 524.72], [700.67, 669.79], [804.99, 453.62], [672.83, 996.0], [771.09, 732.13], [828.69, 779.3]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[807.18, 504.45], [641.27, 662.41], [767.05, 446.88], [752.47, 1016.21], [705.45, 812.7], [908.02, 756.11]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[741.97, 551.99], [565.66, 609.91], [729.11, 440.14], [813.36, 1017.68], [659.52, 927.45], [987.34, 732.91]]) <= 50.0"
] | ballsim | G_3_187 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
160,
57,
199
],
"density": 1,
"position": [
705,
710
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-884.02,
-194.06
]
},
{
"color": [
68,
70,
155
],
"density": 1,
"position": [
686,
965
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
668.54,
714.37
]
},
{
"color": [
94,
240,
80
],
"density": 1,
"position": [
789,
836
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
761.65,
449.68
]
},
{
"color": [
88,
168,
107
],
"density": 1,
"position": [
773,
699
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-273.08,
-930.76
]
},
{
"color": [
217,
155,
153
],
"density": 1,
"position": [
667,
593
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
211.87,
-804.12
]
},
{
"color": [
125,
216,
176
],
"density": 1,
"position": [
594,
569
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
925.42,
-360.52
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 187,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 438809
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (737, 765), initial velocity (-906.62, -54.04) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (883, 767), initial velocity (-907.11, -9.62) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (780, 735), initial velocity (305.79, 904.94) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (788, 898), initial velocity (-666.06, 737.64) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (865, 721), initial velocity (539.63, 686.06) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[646.33, 759.6], [816.76, 859.34], [786.11, 801.02], [721.14, 971.31], [918.96, 736.62]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[609.37, 754.21], [721.69, 880.38], [749.64, 824.46], [623.95, 992.11], [972.91, 744.71]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[700.27, 748.8], [584.19, 828.37], [713.07, 847.96], [618.5, 1012.97], [1027.0, 752.81]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[790.94, 743.4], [717.98, 776.49], [676.6, 871.4], [664.58, 945.26], [1073.54, 748.04]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[895.08, 724.54], [841.67, 738.08], [676.75, 831.35], [638.72, 948.42], [1035.04, 768.97]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[994.31, 738.36], [932.35, 732.69], [693.7, 762.21], [587.01, 996.37], [981.08, 777.05]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[1018.17, 772.15], [985.78, 690.05], [710.64, 693.07], [602.46, 1031.75], [903.43, 808.82]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[1044.79, 736.38], [850.4, 723.38], [727.58, 623.93], [586.78, 963.0], [823.8, 842.59]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[1004.31, 758.3], [714.66, 757.34], [744.53, 554.79], [573.91, 894.25], [744.16, 876.35]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[952.84, 799.24], [586.11, 791.3], [696.98, 597.37], [589.58, 825.5], [664.52, 910.12]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[901.38, 840.19], [733.01, 805.91], [645.59, 646.61], [588.06, 786.55], [584.88, 943.88]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[849.91, 881.13], [880.13, 820.14], [594.2, 695.85], [586.2, 748.17], [659.78, 977.65]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[795.53, 916.69], [985.57, 762.79], [591.21, 691.48], [608.22, 792.24], [685.78, 918.78]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[734.35, 940.78], [963.14, 773.16], [602.95, 672.07], [640.98, 842.15], [711.42, 841.64]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[673.17, 964.88], [859.57, 862.81], [614.69, 652.66], [581.9, 828.21], [792.38, 796.43]]) <= 50.0"
] | ballsim | G_3_364 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
213,
221,
176
],
"density": 1,
"position": [
737,
765
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-906.62,
-54.04
]
},
{
"color": [
121,
90,
145
],
"density": 1,
"position": [
883,
767
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-907.11,
-9.62
]
},
{
"color": [
158,
114,
220
],
"density": 1,
"position": [
780,
735
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
305.79,
904.94
]
},
{
"color": [
158,
158,
125
],
"density": 1,
"position": [
788,
898
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-666.06,
737.64
]
},
{
"color": [
130,
154,
220
],
"density": 1,
"position": [
865,
721
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
539.63,
686.06
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 364,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 434496
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (649, 557), initial velocity (832.79, 3.48) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (805, 866), initial velocity (678.29, 721.73) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (636, 772), initial velocity (-810.59, 306.84) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (564, 607), initial velocity (-833.99, 64.28) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (700, 842), initial velocity (883.47, 323.4) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[732.28, 557.34], [872.82, 938.17], [554.95, 802.69], [538.91, 671.73], [788.35, 874.35]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[815.55, 557.68], [853.47, 952.3], [485.71, 821.56], [532.49, 755.13], [862.68, 863.53]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[899.04, 558.03], [825.98, 1022.42], [443.02, 776.87], [569.68, 813.55], [906.37, 757.83]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[908.15, 632.53], [757.42, 1051.19], [476.84, 714.13], [629.27, 858.84], [949.95, 652.38]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[908.49, 715.81], [731.55, 1034.51], [578.21, 718.94], [688.87, 904.13], [896.72, 643.74]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[908.84, 799.08], [764.81, 982.54], [679.57, 723.74], [743.12, 932.96], [791.28, 687.32]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[909.18, 882.36], [818.46, 1018.99], [730.59, 815.78], [767.41, 869.58], [713.35, 683.24]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[884.15, 940.26], [769.68, 952.99], [654.79, 922.88], [836.0, 831.77], [656.32, 642.99]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[800.87, 939.92], [720.9, 887.0], [666.23, 898.8], [918.51, 802.01], [599.29, 602.73]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[717.6, 939.58], [757.85, 821.01], [600.36, 795.13], [1001.03, 772.25], [590.48, 610.69]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[634.32, 939.24], [807.85, 755.02], [510.59, 691.47], [1083.55, 742.49], [630.74, 667.72]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[591.85, 898.09], [857.85, 689.03], [564.52, 731.52], [1029.26, 787.26], [671.0, 724.75]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[591.51, 815.03], [907.72, 623.2], [667.92, 821.07], [946.94, 816.95], [711.16, 781.64]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[591.16, 731.75], [907.0, 607.94], [771.59, 910.84], [864.43, 846.71], [751.42, 838.67]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[590.82, 648.47], [841.01, 657.94], [854.38, 979.74], [809.62, 860.47], [761.38, 913.2]]) <= 50.0"
] | ballsim | G_3_571 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
178,
73,
244
],
"density": 1,
"position": [
649,
557
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
832.79,
3.48
]
},
{
"color": [
200,
56,
239
],
"density": 1,
"position": [
805,
866
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
678.29,
721.73
]
},
{
"color": [
196,
129,
206
],
"density": 1,
"position": [
636,
772
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-810.59,
306.84
]
},
{
"color": [
122,
132,
207
],
"density": 1,
"position": [
564,
607
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-833.99,
64.28
]
},
{
"color": [
116,
237,
71
],
"density": 1,
"position": [
700,
842
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
883.47,
323.4
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 571,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 821487
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (803, 697), initial velocity (-303.67, 836.04) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (744, 758), initial velocity (138.7, -788.63) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (708, 845), initial velocity (784.78, 486.27) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (649, 726), initial velocity (603.48, 688.62) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (626, 784), initial velocity (-389.4, -815.69) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[772.63, 780.59], [757.87, 679.14], [786.47, 893.63], [709.35, 794.87], [587.06, 702.43]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[769.0, 837.46], [771.73, 600.29], [818.45, 861.75], [728.53, 904.9], [616.9, 620.87]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[780.19, 879.66], [739.87, 600.51], [815.56, 769.23], [665.52, 935.12], [655.94, 539.09]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[791.34, 921.75], [678.52, 651.95], [812.68, 676.94], [596.3, 871.84], [639.32, 553.76]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[761.25, 892.35], [617.17, 703.39], [809.8, 584.66], [713.1, 808.56], [588.15, 628.27]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[730.39, 861.64], [609.21, 754.83], [732.85, 620.69], [829.89, 745.28], [628.06, 702.78]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[699.53, 830.93], [669.37, 807.45], [651.51, 664.34], [946.69, 682.0], [680.42, 776.11]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[749.04, 880.61], [633.72, 795.14], [574.86, 708.0], [970.53, 779.71], [748.21, 834.0]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[785.82, 903.24], [591.27, 776.03], [656.21, 751.65], [898.82, 782.07], [816.01, 891.88]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[742.67, 848.61], [616.21, 756.91], [737.56, 795.3], [785.62, 712.57], [844.68, 820.77]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[698.74, 794.76], [658.66, 737.79], [818.91, 838.95], [672.41, 643.06], [860.11, 724.63]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[654.82, 740.9], [701.12, 718.68], [887.78, 860.99], [585.82, 573.55], [875.54, 628.49]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[611.01, 687.18], [743.47, 699.61], [890.65, 768.94], [689.01, 521.08], [802.36, 686.01]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[597.93, 633.32], [702.9, 763.52], [893.53, 676.66], [685.42, 653.87], [809.85, 664.4]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[641.85, 579.46], [646.09, 843.67], [857.26, 652.19], [681.83, 786.65], [833.58, 626.56]]) <= 50.0"
] | ballsim | G_3_331 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
116,
178,
158
],
"density": 1,
"position": [
803,
697
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-303.67,
836.04
]
},
{
"color": [
97,
125,
114
],
"density": 1,
"position": [
744,
758
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
138.7,
-788.63
]
},
{
"color": [
250,
88,
214
],
"density": 1,
"position": [
708,
845
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
784.78,
486.27
]
},
{
"color": [
141,
231,
131
],
"density": 1,
"position": [
649,
726
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
603.48,
688.62
]
},
{
"color": [
219,
172,
65
],
"density": 1,
"position": [
626,
784
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-389.4,
-815.69
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 331,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4640556
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (690, 753), initial velocity (-618.62, -657.77) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (568, 682), initial velocity (833.46, 398.29) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (987, 686), initial velocity (511.75, 641.53) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (940, 747), initial velocity (-697.26, 520.51) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (952, 580), initial velocity (543.78, 766.86) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[687.77, 721.66], [586.18, 684.21], [1038.17, 750.16], [870.27, 799.05], [1006.37, 656.68]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[773.68, 741.22], [508.03, 630.81], [1048.66, 784.75], [800.55, 851.1], [1060.76, 733.37]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[859.81, 760.82], [495.92, 577.29], [971.64, 755.82], [730.65, 903.28], [1041.12, 756.37]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[915.67, 797.72], [574.06, 523.89], [954.55, 691.65], [660.92, 955.33], [951.56, 728.65]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[934.18, 856.19], [657.69, 487.34], [971.49, 605.38], [595.32, 994.7], [863.74, 703.94]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[952.69, 914.67], [752.29, 484.62], [847.01, 621.89], [569.51, 911.59], [775.92, 679.24]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[930.73, 943.74], [846.89, 481.9], [722.52, 638.39], [543.7, 828.49], [688.1, 654.53]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[869.41, 944.2], [897.93, 510.82], [641.04, 654.89], [517.9, 745.38], [578.76, 629.82]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[808.08, 944.67], [866.1, 599.96], [565.43, 671.4], [492.08, 662.27], [466.77, 605.12]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[746.75, 945.13], [834.26, 689.1], [489.83, 687.9], [466.27, 579.17], [579.05, 580.41]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[685.42, 945.6], [802.43, 778.23], [499.03, 704.41], [507.51, 564.83], [691.33, 555.7]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[624.09, 946.06], [770.59, 867.37], [574.64, 720.91], [577.24, 616.88], [803.61, 530.99]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[562.92, 946.52], [738.83, 956.28], [650.06, 737.37], [646.79, 668.8], [915.61, 506.35]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[501.59, 946.99], [711.36, 1032.0], [725.27, 755.1], [716.73, 720.18], [870.21, 596.21]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[493.01, 947.45], [737.99, 941.18], [786.62, 815.42], [794.25, 748.28], [812.04, 695.35]]) <= 50.0"
] | ballsim | G_3_154 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
77,
160,
124
],
"density": 1,
"position": [
690,
753
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-618.62,
-657.77
]
},
{
"color": [
232,
131,
69
],
"density": 1,
"position": [
568,
682
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
833.46,
398.29
]
},
{
"color": [
114,
54,
96
],
"density": 1,
"position": [
987,
686
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
511.75,
641.53
]
},
{
"color": [
150,
124,
59
],
"density": 1,
"position": [
940,
747
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-697.26,
520.51
]
},
{
"color": [
175,
207,
207
],
"density": 1,
"position": [
952,
580
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
543.78,
766.86
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 154,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 1880051
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (757, 779), initial velocity (573.8, 582.41) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (709, 665), initial velocity (-759.78, -561.27) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (808, 706), initial velocity (-962.68, 13.06) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (908, 749), initial velocity (-548.08, -593.6) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (669, 808), initial velocity (-555.69, 650.29) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[814.37, 837.25], [633.02, 608.87], [711.74, 707.29], [853.19, 689.65], [613.43, 873.04]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[871.75, 895.5], [599.59, 595.29], [615.47, 708.59], [798.38, 630.3], [591.03, 904.91]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[893.76, 918.4], [655.86, 671.45], [518.97, 709.89], [743.43, 570.8], [656.24, 849.21]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[835.51, 861.03], [711.99, 747.43], [433.83, 722.31], [688.62, 511.45], [721.27, 793.64]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[826.22, 839.23], [707.91, 719.15], [463.59, 787.52], [692.92, 511.2], [765.7, 756.17]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[900.53, 878.17], [686.9, 672.52], [559.86, 786.22], [752.27, 566.01], [728.0, 662.35]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[930.56, 872.81], [637.97, 625.89], [656.12, 784.93], [811.62, 620.82], [705.56, 568.52]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[891.62, 798.5], [589.05, 579.26], [752.39, 783.64], [870.97, 675.63], [683.12, 474.7]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[852.68, 724.18], [619.71, 612.21], [848.65, 782.34], [930.32, 730.44], [772.07, 492.26]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[813.74, 649.87], [666.34, 661.14], [944.92, 781.05], [989.67, 785.25], [864.48, 516.11]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[774.8, 575.55], [712.97, 710.06], [1021.15, 759.72], [1029.0, 820.02], [886.92, 609.93]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[735.86, 501.23], [759.6, 758.99], [1008.85, 657.86], [983.78, 770.27], [909.36, 703.76]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[725.24, 455.32], [806.12, 807.79], [903.26, 649.58], [938.68, 720.64], [931.74, 797.35]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[799.56, 494.26], [852.75, 856.72], [797.4, 641.28], [893.47, 670.88], [954.17, 891.17]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[873.88, 533.2], [807.84, 905.64], [691.54, 632.98], [848.25, 621.13], [914.81, 873.18]]) <= 50.0"
] | ballsim | G_3_92 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
64,
228,
254
],
"density": 1,
"position": [
757,
779
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
573.8,
582.41
]
},
{
"color": [
108,
143,
93
],
"density": 1,
"position": [
709,
665
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-759.78,
-561.27
]
},
{
"color": [
148,
132,
222
],
"density": 1,
"position": [
808,
706
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-962.68,
13.06
]
},
{
"color": [
192,
132,
50
],
"density": 1,
"position": [
908,
749
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-548.08,
-593.6
]
},
{
"color": [
237,
251,
233
],
"density": 1,
"position": [
669,
808
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-555.69,
650.29
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 92,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2201246
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (699, 668), initial velocity (-896.2, 442.61) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (737, 713), initial velocity (-934.7, -351.35) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (694, 899), initial velocity (-801.18, 59.8) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (769, 847), initial velocity (885.94, 88.9) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (908, 863), initial velocity (-717.07, -369.02) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (747, 898), initial velocity (-915.19, -398.86) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[607.41, 712.26], [645.66, 677.87], [613.87, 904.98], [821.34, 855.89], [902.66, 826.09], [655.47, 858.11]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[514.1, 756.53], [556.21, 642.74], [563.91, 880.8], [796.65, 864.77], [1038.28, 789.17], [563.94, 818.21]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[441.77, 761.61], [545.43, 686.42], [569.91, 800.47], [771.91, 873.68], [1035.71, 747.83], [483.66, 784.85]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[529.47, 745.01], [552.62, 689.86], [603.83, 806.36], [747.23, 882.57], [900.09, 784.75], [470.84, 790.41]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[668.22, 779.46], [535.58, 617.81], [662.43, 888.2], [722.55, 891.46], [764.47, 821.66], [458.02, 795.98]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[806.96, 813.91], [608.06, 635.29], [709.72, 978.25], [709.16, 892.13], [628.85, 858.58], [453.89, 792.86]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[945.71, 848.35], [680.55, 652.77], [748.49, 1074.49], [704.3, 886.61], [547.83, 840.9], [459.44, 780.04]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[964.86, 763.21], [753.03, 670.25], [715.47, 1026.11], [699.45, 881.1], [584.74, 705.28], [464.99, 767.22]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[930.42, 624.47], [825.52, 687.73], [676.7, 929.87], [694.59, 875.58], [621.65, 569.66], [470.55, 754.41]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[833.39, 548.3], [898.0, 705.21], [625.23, 842.87], [702.43, 860.84], [718.3, 493.77], [476.1, 741.59]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[694.64, 513.85], [970.49, 722.69], [565.51, 761.86], [718.52, 840.11], [811.85, 498.93], [481.66, 728.77]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[672.93, 596.44], [1042.98, 740.17], [509.03, 678.99], [734.61, 819.38], [774.94, 634.55], [484.25, 717.66]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[707.29, 734.83], [1084.26, 742.39], [566.46, 683.25], [750.66, 798.71], [738.12, 769.83], [471.29, 715.57]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[642.18, 816.09], [1011.76, 724.91], [659.18, 722.68], [837.1, 899.82], [771.58, 797.4], [458.28, 713.47]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[569.62, 893.04], [939.27, 707.43], [751.9, 762.11], [839.73, 920.99], [810.3, 816.88], [445.28, 711.37]]) <= 50.0"
] | ballsim | G_3_876 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
107,
157,
77
],
"density": 1,
"position": [
699,
668
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-896.2,
442.61
]
},
{
"color": [
139,
129,
130
],
"density": 1,
"position": [
737,
713
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-934.7,
-351.35
]
},
{
"color": [
118,
139,
178
],
"density": 1,
"position": [
694,
899
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-801.18,
59.8
]
},
{
"color": [
131,
197,
249
],
"density": 1,
"position": [
769,
847
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
885.94,
88.9
]
},
{
"color": [
79,
232,
168
],
"density": 1,
"position": [
908,
863
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-717.07,
-369.02
]
},
{
"color": [
124,
217,
227
],
"density": 1,
"position": [
747,
898
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-915.19,
-398.86
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 876,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3162404
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (774, 663), initial velocity (-827.37, 517.93) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (801, 715), initial velocity (-148.24, 965.79) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (579, 806), initial velocity (911.06, 20.67) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (652, 1000), initial velocity (857.26, 497.1) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (756, 728), initial velocity (269.71, 881.72) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[691.26, 714.78], [808.45, 798.71], [670.11, 808.08], [740.83, 1040.15], [754.02, 832.9]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[608.52, 766.56], [824.05, 877.72], [761.23, 810.15], [839.41, 1050.3], [741.45, 943.91]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[525.58, 818.48], [839.69, 956.92], [852.57, 812.23], [857.84, 1002.07], [736.53, 1031.53]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[490.42, 870.26], [855.29, 876.82], [943.68, 814.31], [817.8, 1070.28], [791.61, 934.34]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[573.16, 922.04], [870.89, 786.22], [1034.79, 816.38], [831.76, 982.78], [846.69, 837.15]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[655.9, 973.82], [886.49, 695.62], [1030.04, 748.8], [845.73, 895.28], [901.77, 739.95]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[738.64, 1025.61], [902.09, 605.02], [999.91, 662.79], [859.69, 807.78], [956.84, 642.76]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[824.45, 1067.95], [917.7, 514.42], [994.92, 602.28], [873.66, 720.28], [918.19, 556.84]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[852.79, 1024.52], [844.44, 488.38], [943.09, 587.63], [887.62, 632.78], [781.88, 526.49]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[816.29, 933.99], [753.45, 475.24], [941.01, 544.26], [851.84, 574.0], [645.58, 496.15]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[779.79, 843.47], [673.0, 494.54], [889.27, 541.62], [784.92, 533.21], [546.57, 580.56]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[743.29, 752.94], [607.11, 558.68], [821.94, 550.32], [718.0, 492.41], [479.12, 685.22]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[706.89, 662.64], [541.38, 622.65], [754.77, 558.99], [670.04, 509.54], [571.32, 789.63]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[670.82, 601.75], [475.49, 686.79], [698.68, 556.43], [628.32, 570.31], [663.75, 894.29]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[656.46, 680.31], [523.65, 750.92], [687.91, 508.56], [524.56, 569.06], [756.18, 998.95]]) <= 50.0"
] | ballsim | G_3_766 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
192,
129,
183
],
"density": 1,
"position": [
774,
663
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-827.37,
517.93
]
},
{
"color": [
162,
152,
186
],
"density": 1,
"position": [
801,
715
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-148.24,
965.79
]
},
{
"color": [
105,
237,
201
],
"density": 1,
"position": [
579,
806
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
911.06,
20.67
]
},
{
"color": [
221,
211,
169
],
"density": 1,
"position": [
652,
1000
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
857.26,
497.1
]
},
{
"color": [
233,
175,
182
],
"density": 1,
"position": [
756,
728
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
269.71,
881.72
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 766,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6076104
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (632, 942), initial velocity (590.37, -623.45) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (784, 681), initial velocity (841.61, -420.5) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (724, 973), initial velocity (-457.74, -870.03) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (682, 777), initial velocity (-245.62, -968.98) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (595, 734), initial velocity (-897.81, -219.35) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[662.49, 879.65], [868.16, 638.96], [735.32, 885.99], [657.44, 680.1], [505.23, 712.08]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[651.65, 817.29], [944.58, 604.64], [829.29, 798.98], [632.88, 583.2], [462.19, 736.88]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[640.78, 754.78], [902.44, 681.07], [923.49, 727.63], [658.97, 536.77], [484.18, 826.88]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[629.94, 692.43], [860.4, 651.11], [986.14, 837.53], [755.87, 561.33], [568.99, 853.77]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[619.1, 630.08], [818.36, 621.16], [844.93, 743.56], [852.77, 585.89], [658.76, 875.69]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[608.26, 567.72], [776.31, 591.2], [703.72, 649.59], [949.66, 610.45], [748.53, 897.62]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[646.96, 554.91], [734.27, 561.24], [596.72, 589.82], [990.0, 691.58], [838.3, 919.54]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[681.41, 581.86], [720.13, 515.18], [690.69, 731.04], [1014.56, 788.48], [913.52, 926.92]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[683.12, 627.71], [738.71, 450.22], [784.66, 872.25], [969.61, 815.87], [891.6, 837.15]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[684.83, 673.56], [742.71, 419.4], [841.53, 976.36], [872.71, 791.31], [869.68, 747.38]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[686.54, 719.41], [724.14, 484.35], [700.32, 882.39], [775.81, 766.75], [847.75, 657.61]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[647.35, 741.65], [705.56, 549.31], [559.11, 788.42], [732.02, 772.85], [825.83, 567.84]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[558.42, 735.05], [687.02, 614.12], [457.65, 734.09], [753.09, 816.32], [803.96, 478.28]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[469.26, 728.43], [668.44, 679.08], [551.62, 875.3], [774.21, 859.9], [743.5, 427.04]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[430.52, 772.23], [649.87, 744.05], [668.85, 993.24], [795.33, 903.48], [747.22, 498.62]]) <= 50.0"
] | ballsim | G_3_104 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
71,
201,
144
],
"density": 1,
"position": [
632,
942
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
590.37,
-623.45
]
},
{
"color": [
106,
135,
110
],
"density": 1,
"position": [
784,
681
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
841.61,
-420.5
]
},
{
"color": [
61,
73,
247
],
"density": 1,
"position": [
724,
973
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-457.74,
-870.03
]
},
{
"color": [
220,
105,
77
],
"density": 1,
"position": [
682,
777
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-245.62,
-968.98
]
},
{
"color": [
79,
162,
50
],
"density": 1,
"position": [
595,
734
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-897.81,
-219.35
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 104,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8477485
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (586, 863), initial velocity (-263.34, 931.79) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (735, 756), initial velocity (-230.51, -971.17) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (841, 884), initial velocity (-654.23, 658.87) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (734, 847), initial velocity (-379.2, -810.32) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (575, 665), initial velocity (182.92, 811.79) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (703, 622), initial velocity (-768.41, -452.06) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[608.52, 907.32], [711.95, 658.88], [775.57, 949.89], [696.08, 765.97], [593.29, 746.18], [626.17, 576.79]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[701.71, 880.98], [688.91, 561.76], [710.14, 1015.79], [658.17, 684.94], [611.57, 827.35], [620.76, 603.0]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[795.13, 854.57], [687.93, 486.53], [726.85, 999.53], [715.97, 699.53], [629.9, 908.73], [592.32, 606.27]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[888.32, 828.23], [785.05, 509.58], [792.75, 934.1], [814.21, 754.69], [648.19, 989.91], [574.04, 619.63]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[981.51, 801.89], [882.17, 532.63], [858.64, 868.67], [912.45, 809.84], [723.42, 1014.14], [602.02, 679.25]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[1072.12, 772.98], [910.66, 624.31], [924.53, 803.24], [989.99, 844.28], [804.6, 1032.44], [630.0, 738.87]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[1027.47, 750.79], [906.75, 674.74], [964.24, 724.45], [978.53, 789.73], [804.26, 969.21], [657.98, 798.49]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[1036.38, 773.03], [793.74, 536.19], [977.92, 671.51], [931.34, 837.38], [785.97, 888.03], [685.96, 858.11]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[1051.68, 766.36], [754.69, 471.61], [949.69, 612.76], [861.25, 907.93], [767.68, 806.85], [713.94, 917.73]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[1028.73, 721.44], [879.93, 561.56], [881.53, 617.55], [791.16, 978.49], [749.4, 725.68], [741.92, 977.35]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[1005.76, 676.52], [804.46, 533.54], [889.15, 753.56], [765.86, 1064.54], [731.11, 644.5], [716.48, 1006.12]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[979.26, 635.12], [691.66, 540.89], [896.77, 889.57], [721.73, 1056.88], [713.54, 564.56], [684.6, 976.25]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[934.45, 612.2], [629.51, 587.87], [829.74, 950.62], [764.95, 1049.23], [710.67, 510.23], [652.79, 879.3]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[889.53, 589.23], [647.63, 715.39], [693.73, 943.01], [808.29, 1041.57], [707.79, 455.76], [620.91, 782.11]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[844.61, 566.25], [665.74, 842.9], [580.4, 912.71], [821.09, 1003.37], [751.04, 447.42], [589.03, 684.92]]) <= 50.0"
] | ballsim | G_3_165 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
98,
128,
246
],
"density": 1,
"position": [
586,
863
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-263.34,
931.79
]
},
{
"color": [
255,
170,
212
],
"density": 1,
"position": [
735,
756
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-230.51,
-971.17
]
},
{
"color": [
176,
134,
166
],
"density": 1,
"position": [
841,
884
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-654.23,
658.87
]
},
{
"color": [
75,
233,
183
],
"density": 1,
"position": [
734,
847
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-379.2,
-810.32
]
},
{
"color": [
240,
84,
174
],
"density": 1,
"position": [
575,
665
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
182.92,
811.79
]
},
{
"color": [
161,
129,
106
],
"density": 1,
"position": [
703,
622
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-768.41,
-452.06
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 165,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2078326
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (705, 647), initial velocity (-866.84, -40.3) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (770, 721), initial velocity (-663.57, -732.13) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (831, 593), initial velocity (788.51, -280.8) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (841, 775), initial velocity (690.77, 522.88) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (672, 874), initial velocity (-523.18, 741.13) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[618.31, 642.97], [703.64, 647.78], [909.86, 564.92], [910.07, 827.29], [619.68, 948.12]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[531.61, 638.94], [637.29, 574.56], [891.83, 633.73], [975.4, 875.85], [667.26, 922.35]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[517.43, 707.63], [643.85, 574.25], [863.69, 712.78], [922.97, 806.61], [741.56, 869.9]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[521.46, 794.33], [717.07, 640.61], [835.61, 791.64], [870.68, 737.54], [815.68, 817.58]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[533.36, 873.15], [790.29, 706.97], [921.28, 756.72], [818.38, 668.48], [776.03, 879.01]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[620.05, 877.18], [863.5, 773.32], [1009.88, 718.88], [766.09, 599.41], [733.45, 943.36]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[706.75, 881.21], [936.72, 839.68], [1036.03, 743.5], [713.79, 530.34], [690.87, 1007.72]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[793.44, 885.24], [948.95, 845.05], [998.19, 832.1], [691.07, 490.84], [717.09, 1003.28]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[880.13, 889.26], [882.59, 771.83], [934.28, 894.64], [760.13, 543.13], [781.44, 960.7]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[876.44, 841.12], [816.23, 698.61], [857.15, 889.46], [829.2, 595.42], [843.37, 920.55]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[881.92, 766.06], [749.87, 625.39], [805.36, 783.77], [898.27, 647.72], [818.96, 966.77]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[896.23, 682.18], [683.52, 552.18], [742.1, 689.56], [967.34, 700.01], [794.54, 1012.99]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[910.5, 598.5], [665.87, 527.69], [679.0, 595.58], [1036.23, 752.17], [770.19, 1059.07]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[862.28, 577.14], [739.09, 594.05], [643.65, 529.27], [1050.52, 749.67], [750.27, 1100.76]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[778.39, 591.44], [812.31, 660.4], [737.86, 592.52], [998.22, 680.6], [778.64, 1058.52]]) <= 50.0"
] | ballsim | G_3_24 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
110,
173,
96
],
"density": 1,
"position": [
705,
647
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-866.84,
-40.3
]
},
{
"color": [
77,
167,
154
],
"density": 1,
"position": [
770,
721
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-663.57,
-732.13
]
},
{
"color": [
253,
75,
58
],
"density": 1,
"position": [
831,
593
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
788.51,
-280.8
]
},
{
"color": [
228,
238,
238
],
"density": 1,
"position": [
841,
775
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
690.77,
522.88
]
},
{
"color": [
175,
139,
88
],
"density": 1,
"position": [
672,
874
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-523.18,
741.13
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 24,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5652958
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (842, 647), initial velocity (-913.09, 402.42) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (779, 702), initial velocity (-644.42, 682.44) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (933, 683), initial velocity (-705.54, 422.9) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (927, 814), initial velocity (323.75, -885.92) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (658, 994), initial velocity (841.06, 416.58) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[750.69, 687.23], [714.55, 770.24], [862.44, 725.29], [959.37, 725.4], [742.11, 1035.65]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[659.38, 727.47], [650.09, 838.47], [791.89, 767.57], [991.75, 636.8], [826.21, 1032.2]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[567.85, 767.8], [585.48, 906.88], [721.15, 809.96], [970.43, 579.03], [910.53, 990.44]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[476.54, 808.04], [526.96, 971.69], [650.6, 852.25], [877.51, 562.77], [988.0, 944.96]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[485.01, 790.66], [618.3, 950.01], [580.04, 894.53], [784.59, 546.51], [982.02, 851.29]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[565.51, 731.7], [709.63, 928.33], [509.48, 936.82], [691.67, 530.25], [976.04, 757.61]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[646.0, 672.74], [800.96, 906.65], [575.91, 900.01], [598.75, 513.99], [970.06, 663.93]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[726.5, 613.78], [892.3, 884.97], [647.81, 860.04], [558.08, 527.89], [964.08, 570.26]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[806.99, 554.82], [983.63, 863.29], [719.71, 820.08], [618.62, 600.23], [941.65, 486.08]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[887.48, 495.86], [1040.67, 821.82], [791.61, 780.11], [679.17, 672.57], [863.52, 456.43]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[902.35, 474.8], [1013.77, 731.88], [863.51, 740.15], [739.72, 744.91], [785.4, 508.45]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[811.04, 515.05], [986.89, 641.94], [935.41, 700.18], [800.27, 817.25], [707.27, 560.45]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[719.96, 555.18], [960.08, 552.22], [1007.13, 660.31], [860.66, 889.41], [629.35, 612.32]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[628.65, 595.42], [933.2, 462.29], [1005.67, 662.71], [921.21, 961.75], [551.22, 664.32]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[537.34, 635.65], [906.32, 518.15], [935.11, 704.99], [920.58, 998.77], [473.1, 716.33]]) <= 50.0"
] | ballsim | G_3_616 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
210,
62,
193
],
"density": 1,
"position": [
842,
647
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-913.09,
402.42
]
},
{
"color": [
100,
106,
125
],
"density": 1,
"position": [
779,
702
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-644.42,
682.44
]
},
{
"color": [
234,
175,
126
],
"density": 1,
"position": [
933,
683
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-705.54,
422.9
]
},
{
"color": [
214,
149,
182
],
"density": 1,
"position": [
927,
814
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
323.75,
-885.92
]
},
{
"color": [
186,
207,
95
],
"density": 1,
"position": [
658,
994
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
841.06,
416.58
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 616,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4266624
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (750, 719), initial velocity (-564.11, -768.43) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (528, 681), initial velocity (-615.83, -557.88) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (751, 794), initial velocity (330.18, 869.04) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (827, 809), initial velocity (665.97, -655.14) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (908, 916), initial velocity (-661.91, 646.2) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (717, 921), initial velocity (-786.36, 581.27) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[693.58, 642.17], [527.12, 685.91], [784.01, 880.91], [893.6, 743.5], [841.81, 980.62], [638.36, 979.13]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[637.16, 565.34], [582.91, 747.48], [817.02, 967.83], [960.2, 677.99], [775.63, 1045.24], [689.59, 907.39]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[664.02, 571.74], [638.83, 809.21], [800.03, 1004.88], [967.31, 671.98], [755.02, 1064.25], [747.87, 828.56]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[740.85, 628.16], [694.62, 870.78], [713.11, 971.87], [901.81, 738.59], [819.65, 998.05], [806.0, 749.92]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[817.68, 684.58], [750.4, 932.35], [626.2, 938.86], [836.31, 805.19], [884.27, 931.86], [864.13, 671.28]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[894.51, 741.0], [806.19, 993.92], [558.19, 886.95], [770.8, 871.79], [948.9, 865.68], [922.26, 592.64]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[971.34, 797.42], [792.17, 985.69], [525.18, 800.04], [705.3, 938.39], [1013.52, 799.49], [861.67, 632.72]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[1001.14, 806.81], [730.6, 929.9], [492.17, 713.12], [657.33, 987.46], [1078.12, 733.3], [783.03, 690.85]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[944.72, 729.98], [669.02, 874.12], [526.12, 693.18], [723.93, 921.96], [1022.08, 787.76], [704.4, 748.98]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[888.3, 653.15], [596.03, 824.92], [613.04, 726.19], [790.53, 856.46], [955.9, 852.38], [637.18, 800.52]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[831.88, 576.32], [472.33, 805.0], [699.95, 759.19], [857.13, 790.96], [889.71, 917.01], [620.65, 822.78]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[775.46, 499.49], [456.04, 714.92], [786.86, 792.2], [923.73, 725.45], [823.52, 981.63], [604.12, 845.05]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[729.51, 433.16], [579.44, 734.79], [873.56, 825.13], [990.17, 660.11], [757.5, 1046.09], [587.63, 867.26]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[806.34, 489.57], [703.14, 754.71], [960.47, 858.14], [949.6, 701.78], [755.71, 1046.29], [571.1, 889.53]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[883.17, 545.99], [826.85, 774.64], [963.84, 807.6], [884.09, 768.39], [820.34, 980.09], [564.13, 902.24]]) <= 50.0"
] | ballsim | G_3_108 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
177,
212,
220
],
"density": 1,
"position": [
750,
719
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-564.11,
-768.43
]
},
{
"color": [
135,
200,
57
],
"density": 1,
"position": [
528,
681
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-615.83,
-557.88
]
},
{
"color": [
160,
241,
72
],
"density": 1,
"position": [
751,
794
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
330.18,
869.04
]
},
{
"color": [
200,
204,
199
],
"density": 1,
"position": [
827,
809
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
665.97,
-655.14
]
},
{
"color": [
105,
247,
237
],
"density": 1,
"position": [
908,
916
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-661.91,
646.2
]
},
{
"color": [
102,
185,
177
],
"density": 1,
"position": [
717,
921
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-786.36,
581.27
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 108,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6766757
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (599, 670), initial velocity (-734.05, 428.73) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (592, 585), initial velocity (648.12, 597.58) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (701, 571), initial velocity (932.0, -44.6) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (992, 820), initial velocity (691.38, -511.93) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (733, 660), initial velocity (-856.99, -352.33) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[525.59, 712.87], [597.71, 644.77], [794.19, 566.53], [1061.14, 768.8], [679.6, 624.77]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[473.43, 755.74], [467.85, 704.53], [887.38, 562.06], [1066.36, 764.04], [700.25, 589.54]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[547.02, 798.72], [575.59, 764.45], [960.37, 572.43], [996.13, 814.09], [720.96, 554.22]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[620.44, 841.59], [705.45, 824.21], [927.34, 659.69], [926.08, 864.02], [741.61, 518.99]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[693.85, 884.46], [835.31, 883.98], [894.3, 746.94], [856.04, 913.95], [762.27, 483.76]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[767.26, 927.33], [914.1, 855.26], [861.27, 834.2], [837.06, 1052.35], [782.92, 448.54]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[840.68, 970.2], [989.73, 821.08], [828.24, 921.45], [894.5, 970.72], [813.77, 444.64]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[913.78, 1012.85], [1065.37, 786.9], [795.21, 1008.71], [966.26, 845.04], [851.19, 461.0]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[850.33, 956.28], [1077.47, 726.63], [785.71, 1023.52], [1038.01, 719.36], [888.62, 477.36]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[786.87, 899.72], [1056.05, 770.69], [810.27, 933.51], [966.19, 593.67], [916.55, 500.62]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[723.42, 843.15], [1017.91, 833.75], [834.83, 843.49], [947.28, 584.39], [859.74, 497.09]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[659.97, 786.58], [979.75, 896.81], [859.39, 753.48], [919.5, 673.38], [768.41, 468.48]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[596.68, 730.15], [941.69, 959.72], [883.89, 663.69], [891.79, 762.14], [695.94, 497.3]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[533.22, 673.59], [903.53, 1022.78], [908.45, 573.67], [864.0, 851.13], [638.86, 574.13]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[469.77, 617.02], [859.64, 1081.65], [904.83, 504.14], [836.22, 940.12], [581.78, 650.97]]) <= 50.0"
] | ballsim | G_3_345 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
61,
197,
231
],
"density": 1,
"position": [
599,
670
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-734.05,
428.73
]
},
{
"color": [
124,
116,
128
],
"density": 1,
"position": [
592,
585
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
648.12,
597.58
]
},
{
"color": [
137,
51,
70
],
"density": 1,
"position": [
701,
571
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
932,
-44.6
]
},
{
"color": [
123,
200,
73
],
"density": 1,
"position": [
992,
820
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
691.38,
-511.93
]
},
{
"color": [
168,
99,
175
],
"density": 1,
"position": [
733,
660
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-856.99,
-352.33
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 345,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4138347
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (780, 678), initial velocity (676.81, 494.46) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (703, 634), initial velocity (661.28, -491.6) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (695, 856), initial velocity (-822.73, 504.32) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (779, 631), initial velocity (-370.22, -763.4) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (728, 727), initial velocity (-577.7, 806.5) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[847.68, 727.44], [731.03, 584.83], [612.72, 906.44], [792.42, 599.31], [670.24, 807.64]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[915.35, 776.88], [721.06, 548.35], [634.57, 956.88], [776.57, 724.53], [612.47, 888.28]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[968.33, 800.69], [677.06, 570.7], [680.74, 944.56], [760.69, 850.06], [610.45, 969.12]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[959.34, 717.36], [633.16, 592.99], [678.21, 848.1], [729.81, 949.25], [619.67, 965.68]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[917.67, 690.64], [589.26, 615.28], [675.67, 751.64], [613.46, 900.37], [586.31, 875.32]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[841.01, 724.53], [619.65, 637.57], [673.13, 655.18], [580.65, 918.79], [583.56, 741.27]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[764.35, 758.41], [663.55, 659.86], [670.59, 558.72], [626.79, 940.15], [586.13, 605.31]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[687.69, 792.3], [707.45, 682.15], [630.0, 528.17], [672.93, 961.51], [590.79, 469.35]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[611.03, 826.19], [751.34, 704.44], [619.84, 574.21], [710.34, 967.76], [682.94, 529.73]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[623.01, 860.07], [795.24, 726.73], [704.65, 620.26], [714.91, 917.12], [798.37, 601.75]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[692.46, 893.96], [839.14, 749.02], [789.47, 666.3], [732.67, 866.49], [913.8, 673.77]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[718.19, 927.85], [883.03, 771.31], [874.28, 712.35], [830.47, 815.85], [1029.23, 745.79]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[731.56, 940.36], [955.57, 793.17], [958.88, 737.19], [870.52, 798.55], [1004.76, 762.98]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[715.08, 901.15], [884.24, 767.44], [979.06, 704.3], [850.28, 816.07], [1048.85, 759.94]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[698.6, 861.94], [801.62, 679.26], [922.98, 760.24], [759.93, 874.08], [1053.81, 754.63]]) <= 50.0"
] | ballsim | G_3_67 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
223,
221,
185
],
"density": 1,
"position": [
780,
678
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
676.81,
494.46
]
},
{
"color": [
59,
130,
126
],
"density": 1,
"position": [
703,
634
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
661.28,
-491.6
]
},
{
"color": [
89,
206,
167
],
"density": 1,
"position": [
695,
856
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-822.73,
504.32
]
},
{
"color": [
149,
89,
67
],
"density": 1,
"position": [
779,
631
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-370.22,
-763.4
]
},
{
"color": [
129,
153,
255
],
"density": 1,
"position": [
728,
727
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-577.7,
806.5
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 67,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4298656
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (830, 713), initial velocity (119.56, 893.57) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (657, 867), initial velocity (-831.62, 29.65) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (744, 781), initial velocity (-74.52, 968.87) m/s\n- Ball 4: regular polygon (4 sides), radius 20.0m, initial position (815, 777), initial velocity (334.18, -864.48) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (796, 629), initial velocity (684.83, -685.39) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[882.48, 677.63], [583.53, 869.95], [736.55, 877.88], [800.24, 838.84], [820.59, 636.49]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[898.57, 693.77], [666.69, 872.91], [722.4, 963.14], [775.22, 931.83], [795.46, 730.07]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[875.59, 777.7], [750.05, 875.87], [634.56, 921.04], [681.88, 906.79], [770.28, 823.88]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[854.29, 856.43], [836.57, 892.45], [578.53, 910.64], [609.27, 861.3], [733.71, 897.63]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[882.78, 781.92], [749.95, 848.63], [610.8, 988.54], [594.01, 758.44], [692.87, 963.97]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[912.37, 704.04], [662.23, 808.17], [599.97, 991.8], [586.27, 655.58], [635.76, 1002.13]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[906.65, 687.31], [582.87, 767.72], [596.39, 924.91], [601.52, 552.73], [587.11, 1004.35]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[853.99, 751.89], [670.59, 727.26], [647.73, 858.01], [593.64, 489.95], [662.78, 1002.77]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[801.33, 816.46], [758.31, 686.81], [699.08, 791.12], [652.82, 554.58], [699.79, 934.24]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[761.26, 859.23], [846.03, 646.36], [750.42, 724.22], [734.27, 619.2], [713.76, 905.61]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[765.16, 825.81], [885.0, 690.32], [775.51, 702.81], [832.76, 654.3], [645.4, 1013.22]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[769.07, 792.39], [893.82, 786.51], [766.91, 739.72], [936.44, 680.41], [628.85, 1015.44]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[785.72, 781.15], [879.51, 842.46], [735.0, 736.1], [994.06, 785.79], [699.58, 916.89]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[776.85, 757.86], [832.7, 810.04], [691.08, 711.81], [948.61, 712.63], [762.27, 803.88]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[720.98, 646.15], [847.15, 822.13], [647.16, 687.52], [866.74, 643.16], [798.85, 771.26]]) <= 50.0"
] | ballsim | G_3_789 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
133,
53,
134
],
"density": 1,
"position": [
830,
713
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
119.56,
893.57
]
},
{
"color": [
240,
117,
159
],
"density": 1,
"position": [
657,
867
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-831.62,
29.65
]
},
{
"color": [
132,
198,
170
],
"density": 1,
"position": [
744,
781
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-74.52,
968.87
]
},
{
"color": [
173,
97,
69
],
"density": 1,
"position": [
815,
777
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
334.18,
-864.48
]
},
{
"color": [
113,
172,
124
],
"density": 1,
"position": [
796,
629
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
684.83,
-685.39
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 789,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 3679575
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (842, 726), initial velocity (209.53, 808.24) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (650, 877), initial velocity (-412.45, -729.27) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (775, 792), initial velocity (667.5, 618.5) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (549, 784), initial velocity (803.16, 71.47) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (885, 899), initial velocity (-24.31, -901.42) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[853.33, 801.28], [608.76, 826.46], [841.75, 853.84], [629.32, 766.69], [901.79, 819.96]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[813.25, 846.86], [567.53, 830.05], [908.5, 915.68], [709.64, 690.22], [1021.44, 800.31]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[773.06, 892.56], [526.19, 833.65], [877.31, 879.58], [790.17, 613.57], [1068.57, 719.38]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[732.97, 938.14], [489.57, 832.61], [815.47, 812.83], [870.49, 537.1], [948.91, 739.03]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[692.88, 983.72], [493.17, 791.38], [753.63, 746.08], [810.42, 601.03], [829.25, 758.68]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[681.64, 1000.46], [496.78, 750.14], [691.79, 679.33], [733.96, 681.35], [709.6, 778.34]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[727.22, 960.37], [500.38, 708.91], [629.95, 612.58], [657.49, 761.67], [589.95, 797.99]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[772.8, 920.28], [503.98, 667.67], [606.5, 584.23], [581.03, 841.99], [470.29, 817.64]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[818.38, 880.2], [525.9, 644.76], [673.25, 646.07], [575.27, 851.61], [489.63, 698.3]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[863.96, 840.11], [589.88, 628.04], [740.0, 707.91], [655.59, 775.15], [528.22, 669.31]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[909.54, 800.02], [674.96, 599.14], [806.74, 769.75], [735.91, 698.68], [560.2, 700.29]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[955.12, 759.93], [760.04, 570.23], [873.49, 831.59], [816.23, 622.22], [592.18, 731.27]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[1000.59, 719.95], [844.91, 541.4], [940.07, 893.28], [895.73, 546.57], [624.09, 762.18]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[1027.52, 698.5], [823.87, 519.49], [899.87, 848.17], [887.93, 666.53], [656.07, 793.16]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[987.44, 744.08], [754.19, 533.96], [838.03, 781.42], [888.62, 791.38], [688.05, 824.14]]) <= 50.0"
] | ballsim | G_3_23 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
225,
162,
133
],
"density": 1,
"position": [
842,
726
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
209.53,
808.24
]
},
{
"color": [
201,
186,
59
],
"density": 1,
"position": [
650,
877
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-412.45,
-729.27
]
},
{
"color": [
164,
52,
145
],
"density": 1,
"position": [
775,
792
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
667.5,
618.5
]
},
{
"color": [
217,
171,
207
],
"density": 1,
"position": [
549,
784
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
803.16,
71.47
]
},
{
"color": [
227,
182,
208
],
"density": 1,
"position": [
885,
899
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-24.31,
-901.42
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 23,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 6581099
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (4 sides), radius 20.0m, initial position (969, 756), initial velocity (591.2, 692.61) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (938, 596), initial velocity (-543.76, 707.87) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (936, 691), initial velocity (-764.06, 387.81) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (727, 517), initial velocity (788.95, 295.86) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (786, 1005), initial velocity (7.71, -929.9) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (729, 750), initial velocity (914.89, 9.18) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[1028.13, 825.26], [883.63, 666.78], [859.58, 729.77], [805.91, 546.59], [786.78, 912.01], [820.48, 750.93]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[988.69, 837.62], [899.79, 667.02], [810.41, 741.32], [884.81, 576.18], [758.17, 840.37], [853.31, 825.22]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[952.48, 874.56], [949.82, 633.44], [747.14, 766.87], [963.92, 605.84], [669.52, 812.13], [901.2, 749.81]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[958.68, 953.73], [973.59, 645.2], [684.03, 792.36], [1010.72, 638.89], [581.09, 783.96], [883.82, 609.43]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[930.03, 1012.78], [893.2, 717.83], [620.92, 817.85], [1021.91, 692.11], [492.66, 755.78], [866.44, 469.05]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[858.35, 1047.0], [812.04, 790.9], [557.81, 843.33], [1033.48, 745.11], [414.25, 733.4], [849.06, 556.48]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[786.67, 1028.27], [730.89, 863.97], [494.7, 868.82], [1045.05, 798.11], [482.87, 795.9], [831.67, 696.86]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[714.99, 994.06], [649.74, 937.05], [501.71, 853.82], [1041.6, 842.44], [551.49, 858.4], [814.29, 837.24]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[643.31, 959.86], [568.59, 1010.12], [555.34, 811.9], [989.9, 858.92], [620.12, 920.9], [796.91, 977.62]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[691.17, 1045.17], [637.35, 996.64], [608.98, 769.99], [938.22, 875.4], [588.2, 882.86], [779.53, 996.86]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[755.25, 981.98], [730.39, 944.17], [662.62, 728.07], [886.53, 891.88], [548.55, 837.08], [762.14, 856.48]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[842.23, 946.52], [788.15, 830.59], [716.26, 686.15], [834.85, 902.82], [508.91, 791.31], [744.76, 716.1]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[929.0, 970.63], [845.77, 717.3], [707.56, 582.13], [783.29, 867.95], [469.34, 745.64], [823.18, 671.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[937.48, 949.48], [933.0, 654.75], [694.45, 473.47], [731.61, 832.99], [429.68, 699.87], [879.1, 583.19]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[873.06, 886.25], [1021.05, 615.59], [681.34, 525.69], [679.92, 798.03], [485.03, 708.93], [924.68, 476.64]]) <= 50.0"
] | ballsim | G_3_149 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
56,
203,
182
],
"density": 1,
"position": [
969,
756
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
591.2,
692.61
]
},
{
"color": [
119,
90,
69
],
"density": 1,
"position": [
938,
596
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-543.76,
707.87
]
},
{
"color": [
101,
242,
233
],
"density": 1,
"position": [
936,
691
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-764.06,
387.81
]
},
{
"color": [
78,
195,
139
],
"density": 1,
"position": [
727,
517
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
788.95,
295.86
]
},
{
"color": [
91,
155,
148
],
"density": 1,
"position": [
786,
1005
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
7.71,
-929.9
]
},
{
"color": [
242,
199,
113
],
"density": 1,
"position": [
729,
750
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
914.89,
9.18
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 149,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4118994
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (948, 549), initial velocity (-345.92, 724.43) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (806, 600), initial velocity (728.03, 599.56) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (637, 811), initial velocity (977.88, 138.3) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (855, 479), initial velocity (186.3, 849.91) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (734, 760), initial velocity (-763.95, 549.66) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[913.41, 621.44], [878.8, 659.96], [681.12, 878.49], [873.63, 563.99], [711.28, 761.3]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[893.43, 683.62], [922.36, 770.56], [671.26, 999.97], [892.26, 633.91], [742.55, 708.62]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[880.31, 768.82], [952.22, 905.34], [661.37, 987.74], [910.93, 669.21], [773.9, 655.8]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[867.23, 853.81], [915.52, 1001.4], [651.51, 866.26], [929.56, 704.41], [805.18, 603.11]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[854.14, 938.79], [784.19, 1042.82], [641.65, 744.77], [948.19, 739.62], [836.45, 550.43]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[841.06, 1023.78], [652.87, 1030.63], [631.78, 623.29], [966.81, 774.82], [867.73, 497.74]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[827.97, 1006.08], [538.9, 979.18], [621.92, 501.8], [985.44, 810.03], [899.0, 445.46]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[814.88, 921.09], [568.69, 844.74], [612.06, 510.18], [1004.07, 845.23], [930.27, 498.16]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[801.8, 836.11], [598.47, 710.29], [602.19, 631.67], [1022.7, 880.44], [961.55, 550.85]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[788.71, 751.12], [507.71, 784.65], [670.62, 617.52], [993.38, 887.95], [992.82, 603.53]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[775.63, 666.14], [530.14, 823.49], [755.82, 574.31], [953.58, 889.41], [1024.1, 656.22]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[762.54, 581.15], [691.59, 782.06], [841.03, 531.1], [913.79, 890.88], [1055.35, 708.9]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[749.49, 496.38], [852.63, 740.74], [926.02, 488.0], [874.09, 892.34], [1086.52, 761.46]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[736.4, 473.75], [1014.08, 699.3], [887.55, 516.2], [834.3, 893.8], [1043.03, 770.99]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[723.31, 558.73], [1010.51, 744.46], [807.52, 568.4], [794.5, 895.27], [968.39, 783.63]]) <= 50.0"
] | ballsim | G_3_275 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
104,
114,
241
],
"density": 1,
"position": [
948,
549
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-345.92,
724.43
]
},
{
"color": [
240,
184,
57
],
"density": 1,
"position": [
806,
600
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
728.03,
599.56
]
},
{
"color": [
105,
82,
78
],
"density": 1,
"position": [
637,
811
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
977.88,
138.3
]
},
{
"color": [
221,
124,
64
],
"density": 1,
"position": [
855,
479
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
186.3,
849.91
]
},
{
"color": [
61,
156,
97
],
"density": 1,
"position": [
734,
760
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-763.95,
549.66
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 275,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5723690
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (767, 589), initial velocity (263.98, 770.56) m/s\n- Ball 2: regular polygon (6 sides), radius 20.0m, initial position (829, 539), initial velocity (537.09, -655.97) m/s\n- Ball 3: regular polygon (5 sides), radius 20.0m, initial position (754, 748), initial velocity (-697.32, -664.99) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (637, 803), initial velocity (946.48, -175.94) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (751, 674), initial velocity (395.31, 703.11) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (533, 745), initial velocity (-913.17, 121.69) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[793.39, 666.05], [821.06, 535.05], [684.27, 681.5], [731.65, 785.4], [790.53, 744.31], [441.68, 757.16]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[820.74, 742.55], [755.46, 588.76], [614.55, 614.99], [794.35, 768.35], [863.93, 814.62], [454.31, 730.68]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[899.99, 789.32], [689.7, 642.6], [604.63, 608.3], [783.63, 781.19], [961.27, 885.11], [545.85, 718.5]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[979.04, 758.65], [671.86, 731.01], [618.95, 640.13], [772.93, 794.01], [894.36, 875.92], [637.16, 706.34]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[1015.6, 710.43], [696.17, 850.06], [587.21, 638.49], [762.24, 806.83], [824.05, 928.94], [728.47, 694.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[924.87, 789.48], [718.84, 968.15], [555.48, 636.86], [751.55, 819.65], [755.54, 983.01], [819.78, 682.02]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[834.15, 868.53], [684.59, 995.68], [523.74, 635.22], [740.85, 832.46], [753.73, 1060.6], [911.09, 669.86]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[787.24, 922.3], [704.89, 950.15], [519.37, 660.95], [730.16, 845.28], [679.91, 987.75], [1002.4, 657.7]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[877.54, 896.86], [700.68, 901.95], [521.0, 692.69], [719.47, 858.1], [617.13, 870.17], [993.21, 746.04]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[967.85, 872.0], [702.53, 905.57], [522.64, 724.43], [708.77, 814.68], [598.38, 706.74], [981.05, 836.76]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[884.35, 845.98], [704.39, 918.38], [524.27, 756.17], [698.08, 762.07], [609.64, 573.32], [968.9, 754.86]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[793.05, 820.54], [706.25, 931.2], [525.91, 787.9], [687.39, 709.46], [773.07, 592.07], [956.74, 664.55]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[701.96, 795.16], [708.1, 943.99], [527.54, 819.56], [676.72, 656.98], [900.07, 631.57], [925.58, 607.42]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[610.66, 769.72], [709.95, 956.81], [529.18, 851.3], [666.03, 604.36], [841.77, 783.86], [849.66, 661.6]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[519.35, 744.28], [711.81, 969.62], [536.0, 877.85], [655.33, 551.75], [783.15, 936.37], [774.03, 715.58]]) <= 50.0"
] | ballsim | G_3_353 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
214,
169,
211
],
"density": 1,
"position": [
767,
589
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
263.98,
770.56
]
},
{
"color": [
78,
59,
144
],
"density": 1,
"position": [
829,
539
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
537.09,
-655.97
]
},
{
"color": [
124,
166,
70
],
"density": 1,
"position": [
754,
748
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-697.32,
-664.99
]
},
{
"color": [
171,
116,
105
],
"density": 1,
"position": [
637,
803
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
946.48,
-175.94
]
},
{
"color": [
253,
201,
107
],
"density": 1,
"position": [
751,
674
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
395.31,
703.11
]
},
{
"color": [
63,
121,
74
],
"density": 1,
"position": [
533,
745
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-913.17,
121.69
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 353,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 4793247
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (747, 742), initial velocity (701.28, 509.1) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (614, 819), initial velocity (857.62, -266.41) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (782, 788), initial velocity (-809.95, -150.65) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (790, 622), initial velocity (-777.1, 256.93) m/s\n- Ball 5: regular polygon (5 sides), radius 20.0m, initial position (759, 843), initial velocity (-28.72, -914.71) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (882, 706), initial velocity (-349.25, 735.36) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[737.35, 713.11], [678.0, 776.55], [854.38, 826.86], [712.29, 647.68], [722.27, 822.96], [863.46, 763.13]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[713.06, 669.58], [666.52, 683.52], [804.41, 833.82], [626.83, 659.94], [750.27, 875.81], [933.73, 731.47]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[708.0, 614.81], [657.36, 643.03], [702.76, 751.51], [646.79, 597.92], [777.25, 926.91], [1003.71, 700.51]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[759.0, 580.49], [636.82, 609.22], [601.37, 669.4], [663.5, 518.58], [745.49, 876.22], [1011.43, 777.19]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[786.44, 622.71], [627.29, 541.54], [651.97, 627.56], [593.14, 590.04], [713.73, 825.54], [964.78, 759.72]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[797.7, 692.98], [585.37, 513.01], [744.66, 651.19], [674.97, 631.71], [681.96, 774.86], [902.23, 714.7]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[808.95, 763.24], [662.68, 568.92], [815.93, 653.42], [722.65, 639.23], [650.2, 724.17], [883.28, 713.27]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[820.21, 833.5], [748.87, 621.57], [783.39, 588.11], [778.54, 652.71], [618.44, 673.49], [954.86, 802.39]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[825.06, 892.67], [750.53, 704.09], [754.4, 562.65], [892.36, 708.27], [586.68, 622.81], [935.85, 734.59]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[769.83, 847.8], [741.79, 797.01], [707.6, 577.08], [829.47, 593.72], [602.47, 572.12], [1007.69, 741.27]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[792.33, 859.39], [648.13, 828.22], [645.77, 591.51], [694.86, 594.08], [634.23, 521.44], [1081.49, 745.97]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[828.8, 881.15], [618.19, 848.34], [644.14, 605.93], [685.94, 594.45], [634.1, 526.02], [1005.83, 778.08]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[842.73, 863.93], [726.85, 868.41], [654.34, 620.33], [844.66, 609.48], [606.16, 578.74], [922.47, 796.44]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[823.02, 840.59], [835.79, 888.52], [664.57, 634.76], [928.79, 754.45], [579.22, 631.6], [863.73, 790.02]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[759.86, 860.71], [877.89, 792.91], [674.8, 649.18], [929.56, 755.02], [607.22, 684.45], [861.44, 727.15]]) <= 50.0"
] | ballsim | G_3_390 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
120,
244,
238
],
"density": 1,
"position": [
747,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
701.28,
509.1
]
},
{
"color": [
225,
110,
244
],
"density": 1,
"position": [
614,
819
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
857.62,
-266.41
]
},
{
"color": [
72,
171,
186
],
"density": 1,
"position": [
782,
788
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-809.95,
-150.65
]
},
{
"color": [
120,
85,
161
],
"density": 1,
"position": [
790,
622
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-777.1,
256.93
]
},
{
"color": [
128,
155,
142
],
"density": 1,
"position": [
759,
843
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-28.72,
-914.71
]
},
{
"color": [
183,
119,
142
],
"density": 1,
"position": [
882,
706
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-349.25,
735.36
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 390,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 708153
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (5 sides), radius 20.0m, initial position (646, 714), initial velocity (34.78, 878.38) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (770, 600), initial velocity (-788.05, -536.89) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (717, 827), initial velocity (-820.81, 2.31) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (770, 669), initial velocity (-871.02, 53.54) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (674, 676), initial velocity (-735.65, -565.18) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (859, 800), initial velocity (-979.99, -35.02) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[651.76, 794.77], [691.19, 546.31], [632.82, 833.69], [682.89, 674.35], [600.44, 619.48], [761.0, 796.51]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[660.12, 835.91], [612.38, 492.63], [631.08, 885.62], [595.78, 679.69], [618.14, 562.96], [687.67, 775.1]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[638.19, 899.18], [585.42, 505.63], [730.2, 937.68], [651.25, 681.99], [687.38, 514.1], [680.26, 719.46]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[616.32, 962.29], [579.06, 600.72], [749.52, 851.81], [666.46, 645.83], [675.22, 606.07], [806.04, 740.82]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[594.44, 1025.4], [586.17, 695.81], [753.99, 740.21], [681.67, 691.69], [663.07, 534.02], [931.82, 762.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[615.19, 1014.66], [593.27, 790.9], [758.46, 628.62], [696.88, 740.96], [622.6, 504.19], [1052.24, 774.26]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[658.89, 964.17], [600.38, 886.0], [722.19, 587.58], [712.09, 790.22], [596.78, 533.08], [1018.23, 790.44]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[702.59, 913.68], [607.48, 981.09], [627.78, 647.25], [727.3, 839.49], [671.15, 561.96], [900.8, 826.28]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[746.29, 927.85], [589.65, 995.05], [631.65, 706.92], [742.51, 829.58], [745.51, 590.84], [775.02, 847.64]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[752.05, 935.48], [668.45, 941.36], [677.56, 738.59], [756.66, 782.89], [819.88, 619.72], [721.59, 843.82]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[753.97, 940.02], [729.99, 875.12], [615.77, 708.07], [877.07, 797.54], [894.24, 648.61], [642.63, 754.15]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[754.55, 910.12], [759.63, 785.69], [601.44, 650.15], [987.37, 794.68], [968.61, 677.49], [658.6, 719.28]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[753.74, 877.9], [789.2, 696.49], [627.03, 569.34], [1042.59, 739.83], [961.95, 706.29], [754.31, 730.56]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[752.94, 845.6], [818.83, 607.06], [642.23, 506.46], [1001.27, 721.43], [880.27, 735.18], [809.96, 741.86]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[752.13, 813.3], [765.93, 660.59], [584.91, 569.18], [921.79, 665.21], [976.21, 764.06], [687.99, 753.16]]) <= 50.0"
] | ballsim | G_3_786 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
199,
219,
165
],
"density": 1,
"position": [
646,
714
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
34.78,
878.38
]
},
{
"color": [
112,
153,
202
],
"density": 1,
"position": [
770,
600
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-788.05,
-536.89
]
},
{
"color": [
208,
192,
118
],
"density": 1,
"position": [
717,
827
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-820.81,
2.31
]
},
{
"color": [
62,
187,
61
],
"density": 1,
"position": [
770,
669
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-871.02,
53.54
]
},
{
"color": [
96,
250,
102
],
"density": 1,
"position": [
674,
676
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-735.65,
-565.18
]
},
{
"color": [
141,
147,
133
],
"density": 1,
"position": [
859,
800
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-979.99,
-35.02
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 786,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5348035
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (878, 571), initial velocity (428.76, -849.16) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (679, 501), initial velocity (910.14, 155.07) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (820, 709), initial velocity (-854.65, -415.23) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (877, 841), initial velocity (-230.2, 798.4) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (745, 820), initial velocity (696.42, 709.92) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[907.5, 495.8], [770.02, 516.5], [734.53, 667.47], [853.98, 920.83], [814.65, 891.0]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[825.16, 490.12], [837.69, 572.44], [649.05, 625.94], [835.93, 1016.02], [879.74, 947.95]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[761.55, 451.63], [867.67, 694.1], [563.37, 584.31], [843.88, 1040.82], [913.06, 978.53]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[730.55, 513.12], [897.58, 815.47], [477.9, 542.78], [889.27, 971.5], [843.34, 914.84]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[701.8, 581.48], [913.68, 894.32], [501.74, 621.58], [942.2, 952.48], [773.61, 826.36]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[673.04, 649.84], [883.3, 830.16], [546.47, 705.42], [882.3, 911.17], [703.88, 737.88]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[644.28, 614.76], [852.93, 766.0], [591.19, 789.26], [801.88, 854.94], [634.16, 752.85]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[615.52, 526.28], [822.56, 701.84], [635.92, 873.1], [721.46, 798.72], [564.43, 821.21]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[626.03, 558.7], [792.19, 637.68], [680.64, 956.93], [641.04, 742.49], [494.71, 889.57]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[654.77, 647.2], [761.82, 573.52], [725.58, 1040.12], [560.62, 686.26], [508.27, 957.92]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[683.5, 735.7], [731.45, 509.36], [811.06, 998.6], [480.2, 630.04], [597.83, 965.2]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[712.24, 824.21], [714.17, 485.5], [896.53, 957.07], [525.84, 573.81], [694.41, 950.87]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[740.9, 912.48], [727.27, 555.08], [981.79, 915.65], [606.06, 517.73], [790.75, 936.58]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[769.64, 1000.99], [740.41, 624.83], [995.22, 821.78], [690.69, 474.47], [887.33, 922.25]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[813.27, 1043.63], [753.54, 694.58], [1008.31, 727.67], [788.78, 472.69], [983.91, 907.92]]) <= 50.0"
] | ballsim | G_3_722 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
255,
158,
105
],
"density": 1,
"position": [
878,
571
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
428.76,
-849.16
]
},
{
"color": [
197,
64,
248
],
"density": 1,
"position": [
679,
501
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
910.14,
155.07
]
},
{
"color": [
238,
161,
67
],
"density": 1,
"position": [
820,
709
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-854.65,
-415.23
]
},
{
"color": [
168,
132,
184
],
"density": 1,
"position": [
877,
841
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-230.2,
798.4
]
},
{
"color": [
95,
200,
191
],
"density": 1,
"position": [
745,
820
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
696.42,
709.92
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 722,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 9589932
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (607, 801), initial velocity (-356.48, 890.76) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (661, 660), initial velocity (778.47, 342.83) m/s\n- Ball 3: regular polygon (3 sides), radius 20.0m, initial position (797, 836), initial velocity (-238.43, -788.96) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (743, 546), initial velocity (-516.49, -689.52) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (770, 750), initial velocity (507.04, 771.06) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[571.36, 890.09], [738.86, 694.28], [734.24, 824.49], [691.34, 477.05], [845.98, 783.33]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[535.71, 979.17], [816.71, 728.55], [651.09, 848.32], [639.68, 480.45], [935.22, 793.7]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[618.34, 1000.14], [894.76, 762.92], [567.73, 872.21], [587.89, 549.57], [1024.67, 804.1]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[713.31, 1013.81], [972.62, 797.2], [484.57, 896.04], [536.23, 618.52], [1044.69, 774.54]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[808.28, 1027.49], [1050.47, 831.47], [538.91, 840.47], [484.58, 687.47], [991.08, 702.47]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[903.25, 1041.16], [993.09, 787.68], [601.12, 780.37], [432.93, 756.41], [937.46, 630.4]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[894.38, 994.88], [924.48, 737.42], [663.32, 720.26], [471.59, 773.22], [883.85, 558.33]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[835.05, 919.47], [855.88, 687.15], [725.53, 660.15], [557.13, 762.96], [830.24, 486.25]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[775.73, 844.05], [787.28, 636.88], [787.74, 600.04], [642.68, 752.71], [776.62, 476.33]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[716.4, 778.36], [718.74, 586.8], [849.94, 539.94], [728.22, 731.83], [722.94, 548.18]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[657.08, 765.22], [684.29, 641.68], [912.15, 479.83], [813.77, 653.54], [628.72, 495.25]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[597.75, 752.09], [649.84, 696.56], [884.29, 471.74], [899.32, 575.24], [591.72, 475.36]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[538.57, 738.99], [615.48, 751.31], [801.34, 495.51], [946.18, 519.36], [684.43, 530.35]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[479.25, 725.85], [581.03, 806.19], [718.19, 519.34], [835.61, 554.29], [777.38, 585.48]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[419.94, 712.72], [546.58, 861.08], [635.03, 543.17], [733.85, 562.08], [859.84, 672.89]]) <= 50.0"
] | ballsim | G_3_212 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
105,
184,
110
],
"density": 1,
"position": [
607,
801
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-356.48,
890.76
]
},
{
"color": [
225,
245,
198
],
"density": 1,
"position": [
661,
660
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
778.47,
342.83
]
},
{
"color": [
239,
169,
89
],
"density": 1,
"position": [
797,
836
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-238.43,
-788.96
]
},
{
"color": [
148,
74,
149
],
"density": 1,
"position": [
743,
546
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-516.49,
-689.52
]
},
{
"color": [
137,
231,
120
],
"density": 1,
"position": [
770,
750
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
507.04,
771.06
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 212,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 425958
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 5 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (776, 803), initial velocity (-893.72, 374.4) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (635, 603), initial velocity (-268.01, 905.35) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (928, 857), initial velocity (-598.62, 797.65) m/s\n- Ball 4: regular polygon (5 sides), radius 20.0m, initial position (534, 821), initial velocity (-65.79, -811.16) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (816, 602), initial velocity (831.04, -319.57) m/s\n- Ball 6: regular polygon (6 sides), radius 20.0m, initial position (868, 531), initial velocity (351.67, -758.62) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[686.62, 840.45], [608.19, 693.53], [868.14, 936.76], [527.43, 739.87], [899.11, 570.04], [883.54, 469.39]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[597.24, 877.9], [581.39, 784.05], [808.27, 1016.52], [520.87, 658.74], [941.38, 567.75], [800.54, 479.38]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[507.64, 915.45], [554.51, 874.81], [776.86, 1008.5], [514.28, 577.41], [885.16, 637.09], [717.32, 489.41]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[494.99, 952.9], [527.71, 965.34], [775.32, 908.79], [525.61, 551.4], [829.08, 706.25], [634.31, 499.4]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[490.07, 931.34], [644.65, 962.67], [773.94, 809.35], [567.97, 620.88], [772.7, 774.88], [555.27, 521.61]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[490.4, 885.29], [770.16, 941.97], [812.05, 778.32], [608.57, 689.09], [640.52, 708.98], [482.22, 562.3]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[490.73, 839.25], [895.68, 921.26], [850.16, 747.29], [522.44, 665.23], [740.3, 811.62], [524.09, 603.0]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[491.06, 793.2], [987.69, 876.22], [888.27, 716.26], [489.31, 641.38], [840.08, 914.26], [597.13, 643.7]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[491.39, 747.16], [968.6, 750.44], [926.38, 685.23], [575.44, 617.53], [902.19, 989.52], [670.18, 684.4]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[491.72, 701.11], [1021.56, 666.26], [928.46, 633.41], [661.58, 593.68], [773.74, 926.34], [743.23, 725.1]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[492.05, 655.07], [964.27, 714.16], [910.62, 570.08], [747.71, 569.82], [645.3, 863.16], [816.27, 765.8]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[492.38, 609.03], [916.08, 761.22], [892.77, 506.75], [833.84, 545.97], [516.86, 799.97], [869.98, 817.66]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[492.7, 563.1], [1005.46, 728.8], [874.97, 443.57], [919.76, 522.18], [524.52, 736.95], [854.83, 909.08]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[503.63, 549.67], [1056.99, 723.96], [840.8, 446.33], [896.24, 577.98], [652.96, 673.76], [839.65, 1000.73]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[530.96, 586.73], [998.4, 799.13], [818.02, 508.05], [846.92, 652.51], [781.4, 610.58], [836.08, 1056.61]]) <= 50.0"
] | ballsim | G_3_543 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
213,
59,
180
],
"density": 1,
"position": [
776,
803
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-893.72,
374.4
]
},
{
"color": [
206,
111,
212
],
"density": 1,
"position": [
635,
603
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-268.01,
905.35
]
},
{
"color": [
194,
153,
54
],
"density": 1,
"position": [
928,
857
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-598.62,
797.65
]
},
{
"color": [
83,
65,
190
],
"density": 1,
"position": [
534,
821
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
-65.79,
-811.16
]
},
{
"color": [
82,
203,
173
],
"density": 1,
"position": [
816,
602
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
831.04,
-319.57
]
},
{
"color": [
156,
246,
192
],
"density": 1,
"position": [
868,
531
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
351.67,
-758.62
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 5
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 543,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 5175424
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 3 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (760, 723), initial velocity (-837.3, -208.5) m/s\n- Ball 2: regular polygon (3 sides), radius 20.0m, initial position (681, 894), initial velocity (-807.02, 486.9) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (690, 807), initial velocity (-272.9, 868.93) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (882, 782), initial velocity (862.63, 259.39) m/s\n- Ball 5: regular polygon (6 sides), radius 20.0m, initial position (829, 796), initial velocity (-474.96, -689.99) m/s\n- Ball 6: regular polygon (3 sides), radius 20.0m, initial position (680, 657), initial velocity (-780.87, 484.36) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[676.26, 702.15], [600.29, 942.68], [662.71, 893.89], [968.25, 807.93], [781.51, 727.01], [601.92, 705.44]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[592.52, 681.3], [614.87, 1009.68], [640.7, 971.62], [1007.13, 751.78], [734.03, 658.01], [621.17, 753.88]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[636.45, 660.4], [628.2, 950.13], [639.98, 1004.09], [991.84, 726.27], [686.43, 588.85], [699.44, 802.43]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[720.19, 639.55], [746.12, 855.89], [609.15, 984.26], [926.24, 788.01], [638.94, 519.85], [777.52, 850.87]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[803.93, 618.7], [825.59, 761.66], [586.71, 964.44], [874.46, 825.82], [585.87, 472.17], [856.39, 881.95]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[864.56, 637.88], [903.66, 667.42], [617.54, 944.61], [910.96, 710.73], [630.82, 532.31], [771.31, 932.44]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[863.51, 702.93], [908.05, 725.98], [648.38, 924.79], [869.55, 766.32], [666.83, 607.94], [686.22, 982.93]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[822.3, 746.15], [908.93, 838.36], [679.21, 904.97], [809.69, 897.57], [702.84, 683.58], [601.14, 1033.42]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[781.09, 789.36], [833.56, 818.65], [687.52, 872.14], [731.0, 944.16], [738.85, 759.21], [577.79, 995.29]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[739.88, 895.8], [846.18, 761.69], [632.88, 802.97], [689.98, 910.63], [720.11, 803.23], [576.59, 896.36]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[662.33, 963.61], [900.9, 704.73], [586.78, 733.81], [630.12, 844.47], [680.31, 835.09], [575.4, 797.44]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[616.53, 933.92], [941.77, 671.75], [641.42, 664.64], [574.77, 778.3], [640.52, 866.95], [574.2, 698.51]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[750.03, 904.3], [919.87, 747.42], [695.92, 595.65], [634.48, 712.31], [600.82, 898.73], [573.01, 599.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[883.87, 874.62], [897.93, 823.27], [724.2, 572.13], [694.34, 646.15], [604.0, 930.59], [573.22, 500.91]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[976.08, 772.84], [834.35, 827.06], [733.43, 581.61], [670.6, 724.8], [643.8, 962.45], [606.72, 464.53]]) <= 50.0"
] | ballsim | G_3_232 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
129,
59,
253
],
"density": 1,
"position": [
760,
723
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-837.3,
-208.5
]
},
{
"color": [
208,
65,
164
],
"density": 1,
"position": [
681,
894
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-807.02,
486.9
]
},
{
"color": [
77,
225,
78
],
"density": 1,
"position": [
690,
807
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-272.9,
868.93
]
},
{
"color": [
234,
212,
114
],
"density": 1,
"position": [
882,
782
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
862.63,
259.39
]
},
{
"color": [
157,
184,
57
],
"density": 1,
"position": [
829,
796
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-474.96,
-689.99
]
},
{
"color": [
161,
104,
98
],
"density": 1,
"position": [
680,
657
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-780.87,
484.36
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 3
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 232,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 8446633
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (774, 736), initial velocity (-849.82, -74.46) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (662, 810), initial velocity (723.02, 562.69) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (975, 756), initial velocity (475.39, 708.75) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (632, 574), initial velocity (-372.36, 803.81) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (883, 848), initial velocity (-2.73, 825.97) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[689.01, 728.55], [734.29, 866.27], [1020.79, 825.13], [594.77, 654.37], [882.73, 930.59]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[604.03, 721.11], [806.58, 922.55], [949.92, 777.6], [557.54, 734.74], [841.8, 972.52]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[518.83, 713.64], [876.02, 975.94], [878.87, 729.94], [520.21, 815.31], [759.0, 972.79]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[446.14, 718.49], [819.75, 903.65], [807.99, 682.41], [548.02, 830.65], [676.41, 973.06]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[455.81, 801.25], [763.47, 831.35], [737.12, 634.88], [628.39, 793.42], [625.66, 941.48]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[540.79, 808.7], [741.03, 778.59], [666.24, 587.34], [682.72, 741.15], [625.93, 858.89]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[596.19, 816.14], [817.74, 783.06], [612.54, 556.98], [660.72, 644.81], [655.79, 776.3]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[596.46, 823.59], [894.45, 787.53], [579.69, 581.44], [719.12, 594.88], [740.77, 693.7]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[596.73, 831.04], [971.16, 792.0], [571.92, 649.64], [821.69, 570.44], [825.76, 611.11]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[597.0, 838.48], [1047.87, 796.46], [572.94, 726.65], [858.91, 576.58], [910.74, 598.05]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[597.27, 845.93], [1054.03, 730.41], [573.97, 803.65], [795.7, 679.15], [948.01, 640.74]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[630.75, 872.56], [1008.69, 694.57], [558.38, 871.06], [732.49, 781.71], [942.98, 725.73]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[662.32, 920.05], [932.17, 690.11], [588.41, 886.66], [674.2, 875.77], [937.96, 810.5]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[735.23, 975.09], [855.46, 685.64], [639.75, 859.23], [662.89, 920.45], [932.93, 895.49]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[882.86, 966.55], [778.75, 681.18], [661.48, 814.7], [681.2, 982.24], [889.73, 900.74]]) <= 50.0"
] | ballsim | G_3_694 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
111,
247,
164
],
"density": 1,
"position": [
774,
736
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-849.82,
-74.46
]
},
{
"color": [
213,
71,
143
],
"density": 1,
"position": [
662,
810
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
723.02,
562.69
]
},
{
"color": [
215,
184,
150
],
"density": 1,
"position": [
975,
756
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
475.39,
708.75
]
},
{
"color": [
103,
191,
115
],
"density": 1,
"position": [
632,
574
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-372.36,
803.81
]
},
{
"color": [
183,
203,
117
],
"density": 1,
"position": [
883,
848
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-2.73,
825.97
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 694,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 20296
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 6 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (6 sides), radius 20.0m, initial position (770, 656), initial velocity (-780.06, -245.0) m/s\n- Ball 2: regular polygon (4 sides), radius 20.0m, initial position (799, 783), initial velocity (-829.84, -446.13) m/s\n- Ball 3: regular polygon (4 sides), radius 20.0m, initial position (814, 1006), initial velocity (958.29, 281.6) m/s\n- Ball 4: regular polygon (3 sides), radius 20.0m, initial position (457, 726), initial velocity (-772.8, 507.06) m/s\n- Ball 5: regular polygon (3 sides), radius 20.0m, initial position (773, 726), initial velocity (852.59, 165.25) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 5\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[692.0, 631.49], [716.02, 738.4], [909.83, 1034.16], [425.79, 750.11], [858.25, 742.53]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[613.99, 606.98], [633.03, 693.79], [884.19, 992.19], [508.34, 708.53], [943.51, 759.06]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[535.8, 582.4], [574.84, 624.07], [811.72, 923.09], [552.61, 705.34], [1028.97, 775.63]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[529.99, 599.58], [555.85, 515.48], [739.43, 854.17], [536.64, 762.27], [1063.86, 763.07]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[590.22, 654.87], [621.23, 455.61], [667.14, 785.25], [520.67, 819.21], [1006.93, 697.5]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[650.45, 710.17], [724.77, 472.74], [594.85, 716.33], [504.71, 876.14], [950.0, 631.92]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[710.68, 765.47], [828.31, 510.58], [522.56, 647.41], [506.7, 922.71], [893.06, 566.34]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[770.91, 820.77], [856.76, 473.32], [515.92, 616.4], [564.0, 937.36], [951.75, 616.39]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[831.14, 876.07], [856.4, 483.25], [611.74, 644.55], [621.3, 952.01], [1054.79, 710.8]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[891.37, 931.36], [856.03, 549.32], [707.57, 672.7], [678.6, 966.66], [1030.75, 731.84]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[951.6, 986.66], [855.66, 615.39], [803.39, 700.85], [735.9, 981.31], [897.45, 689.8]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[898.73, 976.66], [892.59, 616.86], [859.39, 768.84], [793.2, 995.96], [768.06, 685.89]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[820.92, 952.21], [954.6, 574.76], [892.56, 859.36], [850.35, 1010.57], [635.19, 714.17]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[742.92, 927.7], [960.79, 564.86], [925.81, 950.11], [907.65, 1025.22], [501.98, 742.51]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[664.91, 903.19], [893.16, 597.57], [938.32, 1000.57], [905.1, 1048.89], [426.19, 737.71]]) <= 50.0"
] | ballsim | G_3_885 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
62,
201,
110
],
"density": 1,
"position": [
770,
656
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
-780.06,
-245
]
},
{
"color": [
244,
139,
230
],
"density": 1,
"position": [
799,
783
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-829.84,
-446.13
]
},
{
"color": [
218,
88,
109
],
"density": 1,
"position": [
814,
1006
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
958.29,
281.6
]
},
{
"color": [
90,
215,
164
],
"density": 1,
"position": [
457,
726
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
-772.8,
507.06
]
},
{
"color": [
126,
185,
181
],
"density": 1,
"position": [
773,
726
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
852.59,
165.25
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 6
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 885,
"multi_object_count": 5,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 7942992
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 4 sides, radius 375.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (3 sides), radius 20.0m, initial position (836, 909), initial velocity (205.29, -907.23) m/s\n- Ball 2: regular polygon (5 sides), radius 20.0m, initial position (804, 790), initial velocity (0.33, -841.57) m/s\n- Ball 3: regular polygon (6 sides), radius 20.0m, initial position (781, 890), initial velocity (695.5, 397.57) m/s\n- Ball 4: regular polygon (6 sides), radius 20.0m, initial position (681, 769), initial velocity (307.4, -943.14) m/s\n- Ball 5: regular polygon (4 sides), radius 20.0m, initial position (859, 788), initial velocity (-796.94, -339.72) m/s\n- Ball 6: regular polygon (4 sides), radius 20.0m, initial position (728, 665), initial velocity (292.9, 824.86) m/s\n\n### Physics\n- no effective gravity (treated as zero).\n\n### Dynamics\n- No additional time-varying mechanisms.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 6\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] | [
"assert check_distance(predict_position(0.10), [[856.53, 818.28], [786.26, 692.93], [850.56, 929.75], [711.74, 674.69], [800.45, 769.39], [757.3, 747.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[877.06, 727.55], [775.69, 598.78], [878.17, 927.55], [728.51, 572.31], [825.17, 831.41], [712.93, 756.32]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[897.65, 636.61], [798.13, 523.47], [890.43, 909.94], [715.07, 452.21], [794.79, 838.44], [655.9, 752.63]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[900.87, 563.19], [817.91, 504.97], [911.11, 900.84], [780.23, 440.93], [707.09, 788.08], [599.02, 748.94]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[810.15, 583.73], [809.95, 643.66], [884.5, 844.44], [732.16, 454.27], [619.4, 737.71], [542.13, 745.25]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[719.42, 604.26], [801.99, 782.36], [857.89, 788.04], [684.72, 468.24], [531.7, 687.35], [485.25, 741.57]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[628.7, 624.79], [794.03, 921.05], [831.28, 731.65], [671.37, 516.31], [508.04, 701.0], [428.38, 737.88]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[537.98, 645.32], [786.07, 1059.76], [804.67, 675.25], [658.01, 564.38], [558.41, 788.7], [418.52, 765.8]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[486.48, 705.08], [724.62, 998.37], [778.06, 618.86], [644.66, 612.46], [608.77, 876.39], [475.4, 769.49]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[465.95, 795.8], [732.58, 859.68], [751.45, 562.46], [631.3, 660.53], [659.14, 964.09], [532.27, 773.17]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[538.87, 793.07], [740.54, 720.98], [724.83, 506.06], [617.95, 708.6], [709.51, 1051.79], [589.16, 776.86]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[610.45, 791.68], [748.5, 582.28], [702.68, 454.11], [604.59, 756.67], [740.12, 1070.49], [658.47, 768.11]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[665.83, 850.26], [779.5, 514.91], [737.83, 415.71], [591.27, 782.65], [689.88, 983.02], [738.01, 748.97]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[721.34, 909.46], [843.02, 547.23], [756.62, 518.86], [577.92, 808.46], [639.51, 895.32], [817.74, 729.78]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[776.86, 968.66], [906.55, 579.56], [751.08, 648.77], [544.47, 822.65], [615.26, 822.71], [897.48, 710.59]]) <= 50.0"
] | ballsim | G_3_211 | 3 | [
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] | {
"balls": [
{
"color": [
210,
64,
180
],
"density": 1,
"position": [
836,
909
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 3,
"velocity": [
205.29,
-907.23
]
},
{
"color": [
76,
60,
134
],
"density": 1,
"position": [
804,
790
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 5,
"velocity": [
0.33,
-841.57
]
},
{
"color": [
53,
199,
213
],
"density": 1,
"position": [
781,
890
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
695.5,
397.57
]
},
{
"color": [
217,
126,
195
],
"density": 1,
"position": [
681,
769
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 6,
"velocity": [
307.4,
-943.14
]
},
{
"color": [
209,
233,
220
],
"density": 1,
"position": [
859,
788
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
-796.94,
-339.72
]
},
{
"color": [
126,
247,
140
],
"density": 1,
"position": [
728,
665
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 4,
"velocity": [
292.9,
824.86
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 750,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 4
}
],
"meta": {
"difficulty": "hard",
"difficulty_config": {
"description": "Hard: 小空间，复杂布局，高密度，频繁碰撞"
},
"difficulty_level": 3,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"index": 211,
"multi_object_count": 6,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "G",
"seed": 2754916
},
"physics": {
"gravity_enabled": false,
"gravity_x": 0,
"gravity_y": 0
}
} | 50 | function |
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