#### Date of Award

8-1-2016

#### Degree Name

Master of Science

#### Department

Mathematics

#### First Advisor

McSorley, John

#### Abstract

A Latin square of order n is an n × n array in which each row and column contains symbols from an n-set, S = {a1,...,an}, exactly once. If two Latin squares L1 and L2 of the same order can be joined such that each of the n^2 ordered pairs (ai,aj) appears exactly once, then L1 and L2 are said to be orthogonal. This project will involve a variation of this idea. We deﬁne orthogonality of two Latin squares Lm and Ln, for m < n, as follows: When we place an m × m Latin square Lm inside an n × n Latin square Ln, in all possible ways, the so obtained m^2 ordered pairs (ai,aj) are always distinct. We ﬁrst investigate the situation when m = 2 and n = p, where p is a prime.

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