Question 167
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.
* 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.