OMA 2020 Problem 6

Let $n\ge 3$ be an integer. Lucas and Matías play a game in a regular $n$ -sided polygon with a vertex marked as a trap. Initially Matías places a token at one vertex of the polygon. In each step, Lucas says a positive integer and Matías moves the token that number of vertices clockwise or counterclockwise, at his choice.

a) Determine all the $n\ge 3$ such that Matías can locate the token and move it in such a way as to never fall into the trap, regardless of the numbers Lucas says. Give the strategy to Matías.

b) Determine all the $n\ge 3$ such that Lucas can force Matías to fall into the trap. Give the strategy to Lucas.

Note. The two players know the value of $n$ and see the polygon.