Problem

Source: USA Winter TST for IMO 2019, Problem 5, and USA Winter TST for EGMO 2019, Problem 6, by Yannick Yao

Tags: combinatorics



Let $n$ be a positive integer. Tasty and Stacy are given a circular necklace with $3n$ sapphire beads and $3n$ turquoise beads, such that no three consecutive beads have the same color. They play a cooperative game where they alternate turns removing three consecutive beads, subject to the following conditions: Tasty must remove three consecutive beads which are turquoise, sapphire, and turquoise, in that order, on each of his turns. Stacy must remove three consecutive beads which are sapphire, turquoise, and sapphire, in that order, on each of her turns. They win if all the beads are removed in $2n$ turns. Prove that if they can win with Tasty going first, they can also win with Stacy going first. Yannick Yao