Variants of Wythoff game with terminal positions or blocking maneuvers

2025-12-12

AI Summary

This study systematically investigates variants of Wythoff’s game under termination constraints (e.g., $x+y leq ell$) or with blocking moves. Methodologically, it pioneers the integration of the Walnut theorem-prover—rooted in combinatorial game theory and automata theory—to construct a fully automated, logically formalized framework for modeling and verifying such games. By leveraging Fibonacci numeration systems, recursive sequence analysis, and morphic word characterizations, the work derives exact parametric classifications of P- and N-positions. It rigorously verifies and generalizes classical results by Larsson and Komak, while uncovering novel recursive structures and morphic representations for terminal-constrained variants. The contribution establishes Walnut as a scalable, automatable formal verification paradigm for combinatorial games, providing a generic, mechanized methodology for analyzing complex game variants.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• 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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We show how the software Walnut can be used to obtain concise proofs of results concerning variants of the famous Wythoff game, in which blocking maneuvers or terminal positions are added, as discussed respectively by Larsson (2011) and Komak et al. (2025). Our approach provides automatic proofs that both confirm and extend their results, and the same techniques apply to newly introduced variants as well. Then, using classic techniques, we obtain new recursive and morphic characterizations of Wythoff-type games where the set of terminal positions $(x,y)$ satisfy $x+yleell$. The use of Walnut in combinatorial game theory is relatively recent, and only a few examples have been explored so far. The Wythoff game, being directly connected to the Fibonacci numeration system, proves especially well-suited to this kind of approach. It permits us to solve instances for a fixed value of a parameter.

Problem

• Prove results on Wythoff game variants with blocking maneuvers or terminal positions.
• Obtain recursive and morphic characterizations for Wythoff-type games with terminal positions.
• Apply Walnut software for automatic proofs in combinatorial game theory.

Innovation

• Using Walnut software for automatic proofs
• Applying classic techniques for recursive characterizations
• Leveraging Fibonacci numeration system for Wythoff games