The game Max-Welter

TL;DR

This paper analyzes the Max-Welter variant of Welter's game, providing solutions for winning strategies, classifying positions by Sprague-Grundy values, and exploring periodicity and game classification within combinatorial game theory.

Contribution

It introduces the Max-Welter game, solves its winning strategy, classifies positions by Sprague-Grundy values, and situates it within the framework of tame and strongly miserable games.

Findings

Solved the winning strategy for Max-Welter.

Described positions with Sprague-Grundy value 1.

Established periodicity results for Sprague-Grundy values.

Abstract

On a semi-infinite strip of squares rightward numbered $0, 1, 2, \ldots$ with at most one coin in each square, in Welter's game, two players alternately move a coin to an empty square on its left. Jumping over other coins is legal. The player who first cannot move loses. We examine a variant of Welter's game, that we call Max-Welter, in which players are allowed to move only the coin furthest to the right. We solve the winning strategy and describe the positions of Sprague-Grundy value 1. We propose two theorems classifying some special cases where calculating the Sprague-Grundy value of a position of size $k$ becomes easier by considering another position of size $k-1$. We establish two results on the periodicity of the Sprague-Grundy values. We then show that the game Max-Welter is classified in a proper subclass of tame games that Gurvich calls strongly miserable.