Question 31
Let $n \ge 2$ be an integer.
Ariane and Bérénice play a game on the set of residue classes modulo $n$. In the beginning, the residue class $1$ is written on a piece of paper. In each move, the player whose turn it is replaces the current residue class $x$ with either $x+1$ or $2x$. The two players alternate with Ariane starting.
Ariane has won if the residue class $0$ is reached during the game. Bérénice has won if she can permanently avoid this outcome.
For each value of $n$, determine which player has a winning strategy.
Ariane and Bérénice play a game on the set of residue classes modulo $n$. In the beginning, the residue class $1$ is written on a piece of paper. In each move, the player whose turn it is replaces the current residue class $x$ with either $x+1$ or $2x$. The two players alternate with Ariane starting.
Ariane has won if the residue class $0$ is reached during the game. Bérénice has won if she can permanently avoid this outcome.
For each value of $n$, determine which player has a winning strategy.