# Computational experiments on latin squares

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.

## Abstract

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.

## Personnel

• Summer 2011: James Figler and Yudhishthir Singh
• Summer 2012: Megan Bryant and Roger Garcia