The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.  
- The Jury chooses the initial order of the contestants in the queue.  
- Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.  
- If contestant $C_{i}$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.  
- If contestant $C_{i}$ has fewer than $i$ other contestants in front of her, the restaurant opens and the process ends.  
Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
Please provide the value when n = 9.

Let's think step by step and output the final answer within \boxed{}.
user