Previous Player's Positions of Impartial Three-Dimensional Chocolate-Bar Games

Discrete and Computational Geometry, Graphs, and Games

Abstract

In this study, we investigate three-dimensional chocolate bar games, which are variants of the game of Chomp. A three-dimensional chocolate bar is a three-dimensional array of cubes in which a bitter cubic box is present in some part of the bar. Two players take turns and cut the bar horizontally or vertically along the grooves. The player who manages to leave the opponent with a single bitter block is the winner. We consider the $\mathcal{P}$-positions of this game, where the $\mathcal{P}$-positions are positions of the game from which the previous player (the player who will play after the next player) can force a win, as long as they play correctly at every stage. We present sufficient conditions for the case when the position $(p,q,r)$ is a $\mathcal{P}$-position if and only if $(p-1) \oplus (q-1) \oplus (r-1)$, where $p, q$, and $r$ are the length, height, and width of the chocolate bar, respectively.