Grundy Numbers of Impartial Chocolate Bar Games
Ryohei Miyadera, Shunsuke Nakamura, Yushi Nakaya

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 , where is is the largest width of the chocolate and 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 . Therefore, it is natural to look for a necessary and…
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Taxonomy
TopicsArtificial Intelligence in Games
