E530 -- Recherches sur une nouvelle espece de quarres magiques

(Investigations on a new type of magic square)


Euler takes the concept of Latin square (an n by n square containing the numbers 1 through n, each of which appears exactly once in each row and in each column of the square) and generalizes it to a Greco-Latin square (essentially 2 Latin squares laid over each other in a special way). The primary question the paper addresses is: what sizes of Greco-Latin squares are possible to construct?

Euler gives hundreds of examples of Latin and Greco-Latin squares and takes many lengthy detours through this paper, asking questions about Latin squares in which the diagonals also satisfy the "Latin square" property. In the end, he argues, but fails to prove rigorously, that no Greco-Latin square of size 4k + 2 can ever be contructed.

Note: Euler was proven to be wrong in 1970. Greco-Latin squares exist for all possible sizes except 2 and 6. For more discussion of this, see Klyve & Stemkoski, referenced below.

According to the records, it was read to the St. Petersburg Academy on March 8, 1779.

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