Question 184
Carine and Cyril play the following game. First, Cyril chooses an integer $n \geqslant 1$. Then, in each cell of a $3 \times 3$ grid (similar to a tic-tac-toe board), he writes an integer.
Then it is Carine's turn. As many times as she wishes, she may perform the following operation: she chooses a cell $\mathbf{c}$, then increases by 1 the value of $\mathbf{c}$ and of its neighbours (that is, the cells sharing a side with $c$). Carine wins the game if she manages to make the 9 integers all congruent modulo $n$.
Which player has a winning strategy?
Then it is Carine's turn. As many times as she wishes, she may perform the following operation: she chooses a cell $\mathbf{c}$, then increases by 1 the value of $\mathbf{c}$ and of its neighbours (that is, the cells sharing a side with $c$). Carine wins the game if she manages to make the 9 integers all congruent modulo $n$.
Which player has a winning strategy?