$m$-Modular Wythoff

TL;DR

This paper introduces $m$-Modular Wythoff's Game, a variant of the classic game where the equality condition is replaced with a modular equivalence, and characterizes its P-positions.

Contribution

It defines a new modular variant of Wythoff's Game and characterizes its P-positions based on the original game's known positions.

Findings

P-positions are a finite subset of the original game

The modular condition generalizes the equality rule

Characterization aids in understanding game strategy

Abstract

We introduce a variant of Wythoff's Game that we call $m$-Modular Wythoff's Game. In the original Wythoff's Game, players can take a positive number of tokens from one pile, or they can take a positive number of tokens from both piles if the number of tokens they take from the first pile is equal to the number of tokens they take from the second. In our variant, we weaken this equality condition to one of equivalence modulo $m$. We characterize the P-positions of our $m$-Modular variant as a finite subset of the P-positions of the known P-positions of the original Wythoff's Game.