Combinatorial Game Theory: Corners on Young Diagrams

Corners is a combinatorial game played on Young diagrams. This paper, in progress with Dr. Eric Gottlieb, characterizes its P-positions: the positions where the player to move loses under optimal play. The proof goes through a chain decomposition of the diagram paired with a prefix-XOR construction over the chains.

The central technical piece is Lemma 4.3, which shows that the Sprague-Grundy value of any position can be read off the prefix-XOR of a canonical decomposition. I wrote Python and Mathematica code to enumerate and verify the classification on 500+ game states, which was useful both as a sanity check and as a way of building intuition before writing the proof.

The project grew out of Partition Games, a multiplayer platform for 12 combinatorial games on integer partitions that I built during the 2025 Rhodes summer research fellowship. We’ve presented the work at the 57th Southeastern International Conference on Combinatorics, Graph Theory and Computing, the Rhodes-Hendrix-Sewanee Symposium, and as a poster at the Rhodes Undergraduate Research Symposium.