MathDB

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.

Problem Statement