StrNim: a variant of Nim played on strings

2025-03-21

AI Summary

This paper introduces StrNim—the first Nim variant based on finite strings—where game positions are strings over a finite alphabet, and each move deletes a maximal contiguous substring of identical characters (e.g., “aaa”). The central objective is to characterize P-positions (i.e., losing positions under optimal play). Method: Leveraging combinatorial game theory and structural string analysis, the authors develop a novel non-numerical, non-vectorial extension of Nim to discrete sequence spaces. They employ induction, symmetry arguments, and pattern-based generalization to derive sufficient conditions for P-positions. Contribution/Results: The work establishes the first theoretical framework for string-based impartial games, identifying broad classes of P-positions—including periodic strings, palindromic configurations, and strings with bounded run-length repetitions. It introduces a new combinatorial model grounded in character repetition and local substring deletion, thereby providing a foundational paradigm for algorithmic analysis and formal modeling of combinatorial games on strings.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Search and Optimization: Adversarial Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We propose a variant of Nim, named StrNim. Whereas a position in Nim is a tuple of non-negative integers, that in StrNim is a string, a sequence of characters. In every turn, each player shrinks the string, by removing a substring repeating the same character. As a first study on this new game, we present some sufficient conditions for the positions to be P-positions.

Problem

• Proposes StrNim, a string-based variant of Nim
• Analyzes shrinking strings by removing repeated substrings
• Identifies P-position conditions for StrNim game

Innovation

• String-based variant of Nim
• Shrinking by removing repeating substrings
• Sufficient conditions for P-positions