Question 22
Let $n \ge 3$ be an integer. Lucas and Matías play on a regular $n$-gon with one vertex marked as a trap. Initially Matías places a token on a vertex. At each step Lucas announces a positive integer, and Matías moves the token that many vertices clockwise or counterclockwise, as he chooses.
a) Determine all $n \ge 3$ for which Matías can place and move the token so that it never reaches the trap, regardless of Lucas's numbers. Give Matías's strategy.
b) Determine all $n \ge 3$ for which Lucas can force Matías into the trap. Give Lucas's strategy.
a) Determine all $n \ge 3$ for which Matías can place and move the token so that it never reaches the trap, regardless of Lucas's numbers. Give Matías's strategy.
b) Determine all $n \ge 3$ for which Lucas can force Matías into the trap. Give Lucas's strategy.