Investigations of Impartial Games With a Pass

TL;DR

This paper explores Nim with a Pass, a variant allowing a one-time turn skip, and introduces a new binary operation to analyze its properties and generalizations, advancing understanding of its optimal strategies.

Contribution

It defines a new binary operation on games and applies it to Nim with a Pass, providing novel insights into its properties and progress on its $\,\mathcal{P}$-positions and strategies.

Findings

Defined a new binary operation on combinatorial games.

Proved novel properties of Nim with a Pass and its generalizations.

Made progress in identifying $\,\mathcal{P}$-positions and optimal strategies.

Abstract

The study of the combinatorial game Nim and its variants is rich and varied, but little is known of the game Nim with a Pass. It is Nim, but once per game a player is permitted to skip their turn but this can only be done if a nonempty pile remains. In this paper we define a new binary operation on games which we use to prove novel properties of Nim with a Pass, as well as games which are generalizations of it. Most importantly, we make small progress on finding the $\mathcal{P}$-positions and optimal strategies of the game.