Additive Subtraction Games

TL;DR

This paper fully characterizes the nim-value structure of additive subtraction games in the primitive quadratic regime, providing a number theoretic solution and linking nim-values to classical P-positions.

Contribution

It offers a complete proof of the closed formula for P-positions in primitive quadratic additive subtraction games, connecting nim-values to classical P-positions.

Findings

Established the full nim-value structure in the primitive quadratic regime.

Proved that each nim-value sequence is a linear shift of classical P-positions.

Confirmed outcome-periodicity without relying on the closed formula.

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 {\mathscr 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 {\em 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 by 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 {\mathscr P}-positions.