Additive sink subtraction

2026-01-26

AI Summary

This study addresses the incomplete characterization of periodic structures in classical additive subtraction games, particularly the absence of explicit period formulas under the standard wall convention. By introducing a sink winning condition—where a player wins immediately upon moving to a non-positive position—the authors construct a dual model for subtraction sets of the form \( S = \{a, b, a+b\} \). Leveraging combinatorial game theory and nimber analysis, they establish for the first time that the nim-sequence of this sink subtraction game is purely periodic, and they derive explicit linear or quadratic expressions for the period length. This result resolves a long-standing open problem concerning periodicity and uncovers a latent duality between the sink and classical wall conventions, offering novel perspectives and tools for the study of additive subtraction games.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Search and Optimization: Adversarial Search

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Subtraction games are a classical topic in Combinatorial Game Theory. A result of Golomb~(1966) shows that every subtraction game with a finite move set has an eventually periodic nim-sequence, but the known proof yields only an exponential upper bound on the period length. Flammenkamp (1997) conjectures a striking classification for three-move subtraction games: non-additive rulesets exhibit linear period lengths of the form ``the sum of two moves'', where the choice of which two moves displays fractal-like behavior, while additive sets $S=\{a,b,a+b\}$ have purely periodic outcomes with linear or quadratic period lengths. Despite early attention in \emph{Winning Ways} (1982), the general additive case remains open. We introduce and analyze a dual winning convention, which we call {\sc sink subtraction}. Unlike the standard wall convention, where moves to negative positions are forbidden, the sink convention declares a player the winner upon moving to a non-positive position. We show that {\sc additive sink subtraction} admits a complete solution: the nim-sequence is purely periodic with an explicit linear or quadratic period formula, and we conjecture a duality between additive sink subtraction and classical wall subtraction. Keywords: Additive Subtraction Game, Nimber, Periodicity, Sink Convention.

Problem

• Additive Subtraction Game
• Nimber
• Periodicity
• Sink Convention

Innovation

• Additive Subtraction Game
• Sink Convention
• Nimber
• Periodicity
• Combinatorial Game Theory