Additive sink subtraction

TL;DR

This paper introduces the sink subtraction variant of additive subtraction games, providing a complete characterization of its periodic nim-sequences and proposing a duality with classical wall subtraction.

Contribution

It defines and analyzes the additive sink subtraction game, establishing explicit formulas for its periodic nim-sequences and conjecturing a duality with wall subtraction.

Findings

Nim-sequences are purely periodic with linear or quadratic periods.

Explicit formulas for the period lengths are derived.

A conjectured duality exists between sink and wall subtraction.

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 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 {\em 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.