Question 30
Let $n \ge 3$ be a natural number. Anna and Bob play the following game on the vertices of a regular $n$-gon: Anna places her token on a vertex of the $n$-gon. Afterwards Bob places his token on another vertex of the $n$-gon. Then, with Anna playing first, they move their tokens alternately as follows for $2n$ rounds: In Anna's turn on the $k$-th round, she moves her token $k$ positions clockwise or anticlockwise. In Bob's turn on the $k$-th round, he moves his token 1 position clockwise or anticlockwise.
If at the end of any person's turn the two tokens are on the same vertex, then Anna wins the game. Otherwise Bob wins. Decide for each value of $n$ which player has a winning strategy.
If at the end of any person's turn the two tokens are on the same vertex, then Anna wins the game. Otherwise Bob wins. Decide for each value of $n$ which player has a winning strategy.