
TL;DR
This paper introduces a new family of Nim games based on token coloring rules, analyzing winning strategies for two-heap variants with black and white tokens, and explores invariant game concepts.
Contribution
It presents novel Nim variants with coloring-based rules and characterizes winning strategies for specific two-heap families, including new invariant game notions.
Findings
Winning strategies for two-heap Nim variants with black and white tokens.
Characterization of heap sizes as functions of irrational and integer multiples.
Introduction of new concepts in invariant games.
Abstract
We present a new family of Nim games where the rules depend on a given `coloring' of the tokens, each token being either black or white. The rules are as in Nim with the restriction that a white token on top of each heap is not allowed. We resolve the winning strategies of two disjoint game families played on two heaps. The heap-sizes with black tokens correspond to the numbers , where is an integer for one of the families and irrational for the other, and where ranges over the positive integers. In the process, new notions of \emph{invariant games} are introduced.
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Taxonomy
TopicsArtificial Intelligence in Games · Advanced Combinatorial Mathematics · Limits and Structures in Graph Theory
