Strategies for Single-Player PuyoPuyo

Abstract

PuyoPuyo is a Tetris type game which is played worldwide. In this paper, we consider if the player has a winning strategy for single-player PuyoPuyo under various conditions. We regard that the player has a winning strategy if and only if he can keep playing the game forever within a gameboard of a constant height. The difficulty of the game changes according to the number of colors used for game pieces and the width of the gameboard. We first show that $\frac{k(k-1)}{2}+1$ is the sufficient width for the player to have a winning strategy on PuyoPuyo with k colors. Next we show that 3w - 4  (w ≥ 3) colors are sufficient to make the player lose on the gameboard of width w. Finally, we show that the player can make puyos disappear at least once on PuyoPuyo with three colors and width two.