The Complexities of Random-Turn Hex, Square, and Triangle Games

Abstract

In random-turn games, players toss a coin to decide who moves. This paper studies the complexities of the algorithms for playing random-turn connection games perfectly on regular tessellations. Our study theoretically shows that there are algorithms playing random-turn Hex, Square and Triangle perfectly in <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>${O(n^{9}\cdot 2.618^{n}),\ O(n^{9}\cdot 2.746^{n})}$</tex></formula> and <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>${O(n^{9}\cdot 3.645^{n})}$</tex></formula> time for each move respectively, where n is the board size. We then implement the perfect-playing algorithm for random-turn Square and measure the actual running time it costs for each move. We then compute and analyze the game lengths on random-turn Square, Hex and Triangle and conjecture that the asymptotic complexity of their game lengths are the same. We finally compare the perfect-playing algorithm with the sampling algorithm by competing against each other, and the numbers of their wins and loses are reported.