A hidden orthogonal Latin square in a work of Euler from 1770 (Q2663627)

The notions of Latin squares and magic squares are considered. It is noted that L. Euler presented a new construction method for magic squares by means of Latin squares to the Saint Petersburg Academy in October 17, 1776. Also, the concept of orthogonal Latin squares was introduced by Euler. This paper is devoted to a consideration of Euler's investigations in this topic. One can note the following authors' description: ``We discuss the early work (see [\textit{L. Euler}, ``Problema algebraicum ob affectiones prorsus singulares memorabile'', Novi commentarii Academiae Scientiarum Petropolitanae 15, 75--106 (1770), reprinted in Opera Omnia, I-6, 287--315, available online at \url{http://eulerarchive.maa.org} as document E407]) of Euler's on a system of quadratic equations in several unknowns and disclose the hidden orthogonal Latin square. We shall then speculate as to how Euler solved this challenging problem. We conclude with a few, mostly historical, remarks.'' Finally, another approach to Euler's magic square of squares is briefly mentioned. Some historical surveys on the history of Latin squares are cited.