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arXiv · 2505.06206

Constructing All Birthday 3 Games as Digraphs

Alexander Clow·Alfie Davies·Neil Anderson McKay
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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.

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Alexander Clow, Alfie Davies, Neil Anderson McKay. 2025-05-09. Constructing All Birthday 3 Games as Digraphs. https://arxiv.org/abs/2505.06206

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