Elsevier

Discrete Applied Mathematics

Volume 243, 10 July 2018, Pages 39-45
Discrete Applied Mathematics

Ordinal sums of impartial games

https://doi.org/10.1016/j.dam.2017.12.020Get rights and content
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Abstract

In an ordinal sum of two combinatorial games G and H, denoted by G:H, a player may move in either G (base) or H (subordinate), with the additional constraint that any move on G completely annihilates the component H. It is well-known that the ordinal sum does not depend on the form of its subordinate, but depends on the form of its base. In this work, we analyze G(G:H) where G and H are impartial forms, observing that the G-values are related to the concept of minimum excluded value of order k. As a case study, we introduce the ruleset oak, a generalization of green hackenbush. By defining the operation gin sum, it is possible to determine the literal forms of the bases in polynomial time.

Keywords

Combinatorial game theory
Gin sum
Impartial games
Minimum excluded value
Normal-play
OAK
Ordinal sum

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