Investigations of Impartial Games With a Pass
Emet Hirsch

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 -positions and optimal strategies of the game.
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Taxonomy
TopicsArtificial Intelligence in Games · Digital Games and Media
