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Partition games

Abstract : We introduce CUT, the class of 2-player partition games. These are NIM type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can perform on a single heap. In normal play, the player with the last move wins, and the famous Sprague-Grundy theory provides a solution. We prove that several rulesets have a periodic or an arithmetic periodic Sprague-Grundy sequence (i.e. they can be partitioned into a finite number of arithmetic progressions of the same common difference). This is achieved directly for some infinite classes of games, and moreover we develop a computational testing condition, demonstrated to solve a variety of additional games. Similar results have previously appeared for various classes of games of take-and-break, for example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and Guy (1982). In this context, our contribution consists of a systematic study of the subclass `break-without-take'.
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Submitted on : Wednesday, May 6, 2020 - 8:07:07 PM
Last modification on : Wednesday, January 26, 2022 - 7:02:35 PM


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  • HAL Id : hal-01723190, version 3
  • ARXIV : 1803.02621


Antoine Dailly, Eric Duchene, Urban Larsson, Gabrielle Paris. Partition games. Discrete Applied Mathematics, Elsevier, 2020, 285, pp.509-525. ⟨hal-01723190v3⟩



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