Scoring Nim

TL;DR

This paper introduces a new scoring variant of Nim that generalizes traditional normal and misère play, analyzing its theoretical properties, optimal strategies, and payoff functions.

Contribution

It proposes a novel scoring Nim variant that unifies normal and misère Nim, providing a comprehensive theoretical analysis of its strategic and payoff structures.

Findings

Unified framework for Nim variants

Optimal strategies derived for scoring Nim

Analysis of payoff functions and game properties

Abstract

Nim is a well-known combinatorial game in which two players alternately remove stones from distinct piles. A player who removes the last stone wins under the normal play rule, while a player loses under the mis\`ere play rule. In this paper, we propose a new variant of Nim with scoring that generalizes both the normal and mis\`ere play versions of Nim as special cases. We study the theoretical aspects of this extended game and analyze its fundamental properties, such as optimal strategies and payoff functions.