A Subtraction Nim with a Pass

TL;DR

This paper studies a subtraction Nim game with a pass move, proving it maintains the reverse-mex property of Grundy numbers, simplifying analysis despite the pass's introduction.

Contribution

It introduces a pass move into subtraction Nim with specific sets and proves the game retains the reverse-mex property of Grundy numbers.

Findings

The game satisfies the reverse-mex property with the pass move.

The pass move does not complicate the Grundy number structure.

The game analysis extends to a class of subtraction Nim games with passes.

Abstract

We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property of Grundy numbers, i.e., G(x)=mex{G(x+s_1), G(x+s_2), G(x+s_3)}, where the mex is taken over successors rather than predecessors. We modify the rule of this subtraction Nim to allow a one-time pass, that is, a passing move usable at most once during the game, unavailable from terminal positions; once used by either player, it becomes unavailable. In classical Nim, the introduction of a pass move complicates the game, and finding a formula that describes the set of P-positions in traditional three-pile Nim with a pass remains an important open question. In the case of subtraction Nim with a pass, however, the introduction of a pass move does not complicate the game. We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.