Comply/Constrain Subtraction

TL;DR

This paper introduces a closed-form formula for Grundy values in a novel subtraction game where players can impose constraints on each other's moves, and explores periodicity in Grundy values for arithmetic sequences.

Contribution

It develops a closed-form expression for Grundy values in a constrained subtraction game and analyzes their periodicity for arithmetic sequences.

Findings

Derived a closed-form formula for Grundy values.

Analyzed periodicity of Grundy values in arithmetic sequences.

Extended game theory understanding for constrained move sets.

Abstract

A comply/constrain game or a game with a Muller twist is a game where the next player is allowed to place constraints on opponent's next move. We develop a closed form formula for the Grundy value of the single-pile subtraction game where the next player may determine whether the previous player has to select a move from the set of some first k natural numbers or its complement. We also investigate the periodicity of Grundy values when the set of legal moves is from a set of finite arithmetic sequences.