Atomic weights and the combinatorial game of Bipass

TL;DR

This paper introduces Bipass, a new combinatorial game involving swapping stones without bypassing, and establishes a connection to atomic weights, providing strategies, bounds, and analyzing game values.

Contribution

It defines Bipass within normal-play combinatorial games and links it to atomic weights, offering explicit strategies and bounds for game values.

Findings

A surjective function from Bipass strips to atomic weights.

Explicit winning strategies for many Bipass positions.

Proof that the game value *2 cannot be expressed as a sum of Bipass.

Abstract

We define an all-small ruleset, Bipass, within the framework of normal-play combinatorial games. A game is played on finite strips of black and white stones. Stones of different colors are swapped provided they do not bypass one of their own kind. We find a surjective function from the strips to integer atomic weights (Berlekamp, Conway and Guy 1982) that measures the number of units in all-small games. This result provides explicit winning strategies for many games, and in cases where it does not, it gives narrow bounds for the canonical form game values. We prove that the game value *2 does not appear as a disjunctive sum of Bipass. Moreover, we find game values for some parametrized families of games, including an infinite number of strips of value *.