A restriction of Euclid

TL;DR

This paper analyzes a variation of the Euclid game called M-Euclid, where the game ends when one number is a multiple of the other, and determines its Sprague-Grundy function to compare with related versions.

Contribution

It provides a solution for the Sprague-Grundy function of M-Euclid and compares it with Euclid and Grossman's variations.

Findings

Solved the Sprague-Grundy function for M-Euclid

Compared the functions across three Euclid variants

Identified differences in game outcomes based on variations

Abstract

Euclid is a well known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entrees are equal. We examine a further variation that we called M-Euclid in which the game stops when one of the entrees is a positive integer multiple of the other. We solve the Sprague-Grundy function for M-Euclid and compare the Sprague-Grundy functions of the three games.