The goal of this project was to study the number of mates that a latin square may have, as a function of the size of the square. The project was completed during Summer 2011 and Summer 2012 as part of the Marshall University Computational Science REU.

We study the number of mates that a latin square may possess as a function of the size of the square. An exhaustive computer search of all squares of sizes 7 and 8 was performed, giving the exact value for the maximum number of mates for squares of these sizes. The squares of size 8 with the maximum number of mates are exactly the Cayley tables of \( \mathbb{Z}_2^3 = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2\). We obtain a new proof that, for every \( k \geq 2\), the latin square obtained from a Cayley table of \(\mathbb{Z}_2^k \) has a mate.

- Summer 2011: James Figler and Yudhishthir Singh
- Summer 2012: Megan Bryant and Roger Garcia
- Advisor: Dr. Carl Mummert

- Preprint: The number of latin squares of sizes 7 and 8 (pdf preprint).

Published in*Congressus Numerantium*v. 217 (2013): Proceedings of the 44th Southeastern International Conference on Combinatorics, Graph Theory, and Computing. - Poster from Summer 2012: pdf
- Slides from CGTC 44: pdf
- Code: tgz

This research was conducted during the 2011 and 2012 Marshall University REU, which was supported by NSF award OCI-1005117 and by Marshall University. Computational experiments were performed with the Big Green computational cluster at Marshall University, which was supported by NSF award EPS-0918949.