Winning strategies for aperiodic subtraction games

TL;DR

This paper presents a quadratic-time algorithm for computing Sprague-Grundy values in a class of subtraction and division games, providing a winning strategy and characterizing P- and N-positions in misère play.

Contribution

It introduces an efficient method to determine winning strategies for aperiodic subtraction games and characterizes position types in misère play, solving a previously open problem.

Findings

Quadratic-time algorithm for Sprague-Grundy values

Characterization of P- and N-positions in misère play

Solution to a problem posed by Fraenkel

Abstract

We provide a winning strategy for sums of games of MARK-t, an impartial game played on the nonnegative integers where each move consists of subtraction by an integer between 1 and t-1 inclusive, or division by t, rounding down when necessary. Our algorithm computes the Sprague-Grundy values for arbitrary n in quadratic time. This solves a problem posed by Aviezri Fraenkel. In addition, we characterize the P-positions and N-positions for the game in mis\`ere play.