Chromatic Nim finds a game for your solution

TL;DR

Chromatic Nim introduces a coloring rule based on an increasing sequence, creating a new variation of Nim with unique winning strategies and solutions for specific sequences, resolving a problem posed in combinatorial game theory.

Contribution

The paper presents explicit characterizations for winning strategies in Chromatic Nim for certain sequences and a general solution based on color dominance, addressing a problem from 2011.

Findings

Explicit solutions for Beatty sequences and evil numbers.

A general approach based on color dominance.

Resolution of a problem posed at BIRS 2011.

Abstract

We play a variation of Nim on stacks of tokens. Take your favorite increasing sequence of positive integers and color the tokens according to the following rule. Each token on a level that corresponds to a number in the sequence is colored red; if the level does not correspond to a number in the sequence, color it green. Now play Nim on a arbitrary number of stacks with the extra rule: if all top tokens are green, then you can make any move you like. On two stacks, we give explicit characterizations for winning the normal play version for some popular sequences, such as Beatty sequences and the evil numbers corresponding to the 0s in the famous Thue-Morse sequence. We also propose a more general solution which depends only on which of the colors `dominates' the sequence. Our construction resolves a problem posed by Fraenkel at the BIRS 2011 workshop in combinatorial games.