Investigations on a new type of magic square
Euler was proven to be wrong in 1970: Graeco-Latin squares exist for all possible sizes except 2 and 6. For further discussion of this, see Klyve & Stemkoski, "Graeco-Latin Squares and a Mistaken Conjecture of Euler," College Math. J., 2006, Vol. 37 (1), pp. 2-15
Euler takes the concept of Latin square (an n×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 Graeco-Latin square (essentially, two Latin squares laid over each other in a special way). The primary question the paper addresses is: what sizes of Graeco-Latin squares are possible to construct? Euler gives hundreds of examples of Latin and Graeco-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 Graeco-Latin square of size 4k+2 can ever be constructed.
Original Source Citation
Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen, Volume 9, pp. 85-239.
Opera Omnia Citation
Series 1, Volume 7, pp.291-392.