Playability and arbitrarily large rat games

TL;DR

This paper constructs combinatorial games based on Fraenkel's sequences, demonstrating their role as winning positions for the second player, and explores their implications for game theory and number sequences.

Contribution

It introduces new playable combinatorial games linked to Fraenkel's sequences, expanding understanding of their structure and potential applications in game theory.

Findings

Sequences form the second player's winning positions

Games are constructed with playable rulesets

Implications for Fraenkel's conjecture and game theory

Abstract

In 1973 Fraenkel discovered interesting sequences which split the positive integers. These sequences became famous, because of a related unsolved conjecture. Here we construct combinatorial games, with `playable' rulesets, with these sequences constituting the winning positions for the second player. Keywords: Combinatorial game, Fraenkel's conjecture, Impartial game, Normal play, Playability, Rational modulus, Splitting sequences