Question 38
Anton and Britta play a game with the set $M=\{1,2,3, \ldots, n-1\}$ where $n \geq 5$ is an odd integer. In each step Anton removes a number from $M$ and puts it in his set $A$, and Britta removes a number from $M$ and puts it in her set $B$ (both $A$ and $B$ are empty to begin with). When $M$ is empty, Anton picks two distinct numbers $x_{1}, x_{2}$ from $A$ and shows them to Britta. Britta then picks two distinct numbers $y_{1}, y_{2}$ from $B$. Britta wins if
$$
\left(x_{1} x_{2}\left(x_{1}-y_{1}\right)\left(x_{2}-y_{2}\right)\right)^{\frac{n-1}{2}} \equiv 1 \quad \bmod n
$$
otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.
$$
\left(x_{1} x_{2}\left(x_{1}-y_{1}\right)\left(x_{2}-y_{2}\right)\right)^{\frac{n-1}{2}} \equiv 1 \quad \bmod n
$$
otherwise Anton wins. Find all $n$ for which Britta has a winning strategy.