Constructing All Birthday 3 Games as Digraphs

TL;DR

This paper demonstrates that all game values born by day 3 in combinatorial game theory can be represented by Digraph Placement games with at most 8 vertices, extending previous bounds and providing new combinatorial proofs.

Contribution

It extends the known bounds for representing game values with Digraph Placement games to day 3, using exhaustive and random searches, and offers improved bounds for days 4 and 5.

Findings

All 1474 values born by day 3 correspond to Digraph Placement games on at most 8 vertices.

The bound of 8 vertices for day 3 is proven to be optimal.

Improved bounds are provided for days 4 and 5.

Abstract

Recently, Clow and McKay proved that the Digraph Placement ruleset is universal for normal play: for all normal play combinatorial games $X$, there is a Digraph Placement game $G$ with $G=X$. Clow and McKay also showed that the 22 game values born by day 2 correspond to Digraph Placement games with at most 4 vertices. This bound is best possible. We extend this work using a combination of exhaustive and random searches to demonstrate all 1474 values born by day 3 correspond to Digraph Placement games on at most 8 vertices. We provide a combinatorial proof that this bound is best possible. We conclude by giving improved bounds on the number of vertices required to construct all game values born by days 4 and 5.