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# Graeco-Latin squares

Let $A=(a_{{ij}})$ and $B=(b_{{ij}})$ be two $n\times n$ matrices. We define their *join* as the matrix whose $(i,j)$th entry is the pair $(a_{{ij}},b_{{ij}})$.

A *Graeco-Latin square* is then the join of two Latin squares.

The name comes from Euler’s use of Greek and Latin letters to differentiate the entries on each array.

An example of Graeco-Latin square:

$\begin{pmatrix}a\alpha&b\beta&c\gamma&d\delta\\ d\gamma&c\delta&b\alpha&a\beta\\ b\delta&a\gamma&d\beta&c\alpha\\ c\beta&d\alpha&a\delta&b\gamma\end{pmatrix}$ |

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## Comments

## Graeco-Latin squares of order 7 and 8

Does anyone know whether there has been a complete search for all Graeco-Latin squares of order 7 or 8? (Not counting isomorphisms)

It is easy to verify that there is one square of order 3, one square of order 4, and two non-isomorphic Graeco-Latin squares of order 5.

Since it has been proven that there are no Graeco-Latin squares of order 6, the natural question is how many squares of order 7 or 8 there are.

Two Graeco-Latin squares are said to be isomorphic if one can be converted to the other through a series of operations: rearranging rows, rearranging columns, transposing the square, permuting the Greek letters, permuting the Latin letters, or exchanging the Greek and Latin lette