A complete solution for the partisan chocolate game

Tomoaki Abuku (), Hikaru Manabe (), Richard J. Nowakowski (), Carlos P. Santos () and Koki Suetsugu ()
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Tomoaki Abuku: National Institute of Informatics
Hikaru Manabe: Keimei Gakuin Junior and Senior High School
Richard J. Nowakowski: Dalhousie University
Carlos P. Santos: Center for Mathematics and Applications (NovaMath), FCT NOVA
Koki Suetsugu: National Institute of Informatics

International Journal of Game Theory, 2024, vol. 53, issue 4, No 13, 1369-1384

Abstract: Abstract The class of Poset Take-Away games—a player chooses an element and it, and everything above it, is removed—includes many interesting and difficult games. Playing on an n-dimensional positive quadrant (the origin being the bottom of the poset) gives rise to nim, wythoff’s nim and chomp. These are impartial games. We introduce a partisan game motivated by chomp and the recent chocolate-bar version. Our game is played on a checkerboard, with the bottom left square being the origin, and where one player can only choose white squares, the other black. Alternately, this can be regarded as a chocolate bar with alternating different flavored squares. We solve this game by showing it is equivalent to blue-red hackenbush strings. This equivalence proves that all the positions in this game are numbers, and allows us to show that all numbers can be formed by disjunctive sums. Given that almost-all games are first player wins, this shows that this ruleset is special. There are many natural generalizations of the basic partisan game, but all considered had first-player win positions.

Keywords: Combinatorial game theory; Chocolate-bar games; Jacobsthal sequence; 91A46; 91A05; 91A80 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00182-024-00919-x

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