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

TL;DR

This paper analyzes three-dimensional chocolate bar games, identifying conditions under which certain positions are winning or losing, extending the understanding of impartial combinatorial games in three dimensions.

Contribution

It introduces sufficient conditions for P-positions in 3D chocolate bar games, generalizing previous two-dimensional results to three dimensions.

Findings

Characterization of P-positions using bitwise XOR conditions

Sufficient conditions for positions to be winning or losing

Extension of Chomp game analysis to three dimensions

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 P-positions of this game, where the 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 P-position if and only if the bitxor of p-1, q-1, r-1, where p, q and r are the length, height, and width of the chocolate bar, respectively.