Multi-Dimensional Chocolate and Nim with a Pass
Ryohei Miyadera, Hikaru Manabe

TL;DR
This paper extends chocolate-bar game theory to multiple dimensions, characterizes certain functions for these games, and explores how introducing a pass move affects the complexity of Nim, addressing a key open question in combinatorial game theory.
Contribution
The paper generalizes chocolate games to higher dimensions and applies this to analyze the impact of a pass move in Nim, revealing new insights into game complexity.
Findings
Characterization of functions F where Sprague--Grundy values follow XOR pattern
Analysis of multi-dimensional chocolate games with pass moves
Insights into the complexity of Nim with a pass
Abstract
Chocolate-bar games are variants of the CHOMP game. A three-dimensional chocolate bar comprises a set of cubic boxes sized 1 X 1 X 1, with a bitter cubic box at the bottom of the column at position (0,0). For non-negative integers u,w such that u < x and w \< z, the height of the column at position (u,w) is min (F(u,w),y) +1, where F is a monotonically increasing function. We denote this chocolate bar as CB(F,x,y,z). Each player, in turn, cuts the bar on a plane that is horizontal or vertical along the grooves, and eats the broken piece. The player who manages to leave the opponent with the single bitter cubic box is the winner. In this study, functions F such that the Sprague--Grundy value of CB(F,x,y,z) is x xor y xor z are characterized. We have already published the research on three-dimensional chocolate games. In this paper, the authors study a multi-dimensional chocolate game,…
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Taxonomy
TopicsArtificial Intelligence in Games
