Additive Subtraction Games

2026-03-11

AI Summary

This study investigates the Sprague–Grundy (nim) value structure of subtraction games in the primitive quadratic case, aiming to verify the closed-form formula for P-positions conjectured in *Winning Ways*. By integrating techniques from number theory, Beatty sequence theory, and combinatorial game theory, the work provides the first complete proof that the P-positions are precisely characterized by a rational-modulus Beatty-type bracket expression. Moreover, it demonstrates that all nim-value sequences lie on linear translates of the set of P-positions, thereby establishing the full nim-value structure in the primitive quadratic setting and elucidating the exact correspondence between P-positions and nim values.

Technology Category

• Game Theory and Economic Paradigms: Behavioral Game Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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
• Security and Privacy: Applications of cryptography

Abstract

We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em bracket expressions} on rational moduli, for determining the P-positions, but to the best of our knowledge, a complete proof of this claim has not yet appeared in the literature; Mikl\'os and Post (2024) established outcome-periodicity, but without reference to that closed formula. The primitive quadratic case captures the source of the quadratic complexity of the problem, a claim supported by recent research in the dual setting of sink subtraction with Bhagat et al. This study focuses on a number theoretic solution involving the classical closed formula, and we establish that each nim-value sequence resides on a linear shift of the classical P-positions.

Problem

• additive subtraction games
• nim-value
• P-positions
• Beatty sequences
• quadratic complexity

Innovation

• additive subtraction games
• nim-values
• Beatty sequences
• P-positions
• primitive quadratic regime