Grundy Numbers of Impartial Chocolate Bar Games

TL;DR

This paper investigates the Grundy numbers of step chocolate bar games, providing necessary and sufficient conditions for when these numbers equal a specific XOR expression, extending previous results based on the proportionality constant.

Contribution

The authors establish precise conditions under which the Grundy number of chocolate bar games equals a specific XOR-based value, generalizing prior findings for different proportionality constants.

Findings

Derived necessary and sufficient conditions for Grundy numbers to equal $(m-1)  (n-1)$

Extended analysis to cases where the Grundy number equals $((m-1)  (n-1+s))-s$

Generalized previous results for proportionality constants that are even or odd

Abstract

Chocolate bar games are variants of the CHOMP game in which the goal is to leave your opponent with the single bitter part of the chocolate. In this paper, we investigate step chocolate bars whose widths are determined by a fixed function of the horizontal distance from the bitter square. When the width of chocolate bar is proportional to the distance from the bitter square and the constant of proportionality is even, the authors have already proved that the Grundy number of this chocolate bar is $(m-1) \oplus (n-1)$, where $m$ is is the largest width of the chocolate and $n$ is the longest horizontal distance from the bitter part. This result was published in a mathematics journal(Integers,15, 2015). On the other hand, if the constant of proportionality is odd, the Grundy number of this chocolate bar is not $(m-1) \oplus (n-1)$. Therefore, it is natural to look for a necessary and sufficient condition for chocolate bars to have the Grundy number that is equal to $(m-1) \oplus (n-1)$, where $m$ is the largest width of the chocolate and $n$ is the longest horizontal distance from the bitter part. In the first part of the present paper, the authors present this necessary and sufficient condition. Next, we modified the condition that the Grundy number that is equal to $(m-1) \oplus (n-1)$, and we studied a necessary and sufficient condition for chocolate bars to have Grundy number that is equal to $((m-1) \oplus (n-1+s))-s$, where $m$ is is the largest width of the chocolate and $n$ is the longest horizontal distance from the bitter part. We present this necessary and sufficient condition in the second part of this paper.