Multi-Dimensional Chocolate and Nim with a Pass

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, where the dimension is more than three, and apply the theory to the problem of pass move in Nim. We modify the standard rules of the game to allow a one-time pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once a pass has been used by either player, it is no longer available. It is well-known that in classical Nim, the introduction of the pass alters the underlying structure of the game, significantly increasing its complexity. A multi-dimensional chocolate game can show a perspective on the complexity of the game of Nim with a pass. Therefore, the authors address a longstanding open question in combinatorial game theory. The authors present this paper, since their discovery seems to be significant. It appears to the authors that the relation between chocolate games and Nim with a pass will be an essential research topic soon.