Impartial games whose rulesets produce given continued fractions

TL;DR

This paper constructs impartial take-away games with rulesets designed to produce P-positions aligned with specific continued fractions, reversing the typical analysis of P-positions to ruleset design.

Contribution

It introduces a method to derive simple, invariant rulesets for impartial games that generate P-positions based on complementary Beatty sequences linked to continued fractions.

Findings

Rulesets produce P-positions matching given continued fractions

Rules are given by closed-form formulas and are invariant

The approach reverses traditional P-position analysis in combinatorial game theory

Abstract

We study 2-player impartial games of the form take-away which produce P-positions (second player winning positions) corresponding to complementary Beatty sequences, given by the continued fractions (1;k,1,k,1,...) and (k+1;k,1,k,1,...). Our problem is the opposite of the main field of research in this area, which is to, given a game, understand its set of P-positions. We are rather given a set of (candidate) P-positions and look for "simple" rules. Our rules satisfy two criteria, they are given by a closed formula and they are invariant, that is, the available moves do not depend on the position played from (for all options with non-negative coordinates).