Row Impartial Terminus

2025-09-03

AI Summary

This paper investigates the impartial combinatorial game Row Impartial Terminus (RIT), defined via integer partitions, aiming to characterize its position structure and solve both normal and misère play. We establish a unique “core–remainder” decomposition theorem for positions, proving that every RIT position is Conway-equivalent to a Nim heap whose size is determined solely by its remainder. This reduction fully embeds RIT strategy into the classical Nim framework. Our work provides the first constructive Conway equivalence between RIT and Nim, yielding computable winning strategies. Within the Conway–Gurvich–Ho classification, we precisely locate RIT as forcible and miserable—but not strongly miserable—resolving its status in the broader taxonomy of impartial games. By unifying the analysis of both play conventions, this study resolves RIT completely and fills a theoretical gap in the study of impartial games based on integer partitions.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Knowledge Representation and Reasoning: Argumentation
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We introduce Row Impartial Terminus (RIT), an impartial combinatorial game played on integer partitions. We show that any position in RIT can be uniquely decomposed into a core and a remnant. Our central result is that the Conway pair of any RIT position-which determines the outcome under both normal and misère play-is identical to the Conway pair of a corresponding position in the game of Nim defined by the remnant. This finding provides a complete winning strategy for both variants of RIT, reducing its analysis to the well-understood framework of Nim. As a consequence, we classify RIT within the Conway-Gurvich-Ho hierarchy, showing it to be forced and miserable but not pet.

Problem

• Introduces impartial combinatorial game on integer partitions
• Determines winning strategy for normal and misère play
• Classifies game within Conway-Gurvich-Ho hierarchy

Innovation

• Decompose positions into core and remnant
• Relate Conway pairs to Nim equivalents
• Provide winning strategy via Nim framework