A number game reconciliation

TL;DR

This paper critically re-examines the theory of number games in combinatorial game theory, identifying inconsistencies and proposing a refined, more coherent framework that clarifies ambiguities and highlights open problems.

Contribution

It offers a new, detailed characterization of number games, addressing conceptual gaps and refining the definitions of numbers within the theory.

Findings

Identified subtle inconsistencies in existing literature.

Developed a refined classification of number games and their subclasses.

Highlighted open problems for future research.

Abstract

Number games play a central role in alternating normal play combinatorial game theory due to their real-number-like properties (Conway 1976). Here we undertake a critical re-examination: we begin with integer and dyadic games and identify subtle inconsistencies and oversights in the established literature (e.g. Siegel 2013), most notably, the lack of distinction between a game being a number and a game being equal to a number. After addressing this, we move to the general theory of number games. We analyze Conway's original definition and a later refinement by Siegel, and highlight conceptual gaps that have largely gone unnoticed. Through a careful dissection of these issues, we propose a more coherent and robust formulation. Specifically, we develop a refined characterization of numbers, via several subclasses, dyadics, canonical forms, their group theoretic closure and zugzwangs, that altogether better capture the essence of number games. This reconciliation not only clarifies existing ambiguities but also uncovers several open problems.