# Latin squares

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## Abstract

In 1782 in Recherches sur une nouvelle espèce de quarrés magiques, Leonhard Euler posed the following problem:

Given $36$ officers of $6$ ranks and from $6$ regiments, can they be arranged in a $6$-by-$6$ formation so that in each row and column there is one officer of each rank and one officer from each regiment?

Euler conjectured that this was not possible, but it was not proven until Gaston Tarry in 1901. In more modern language, Euler’s question is equivalent to asking whether or not there exist two orthogonal Latin squares of order $6$. In fact, Euler conjectured that no pair of orthogonal Latin squares of order $n$ exists, where $n = 6,10,14,\dots,4k+2,\ldots.$ However, this conjecture was shown to be false for $n>6$ by R. C. Bose, S. S. Shrikhande, and E. T. Parker in 1960. In this talk, we will define the notion of a Latin square of order $n$ and what it means for a pair of Latin squares to be orthogonal, describe ways to construct Latin squares of order $n$, and describe bounds on the largest number of orthogonal Latin squares of order $n$.

## Note

This talk is very similar to the following: