Summary: In 1779 L. Euler [in Opera omnia, Ser. I, Vol. VII, 291–392, Teubner, Leipzig (1923; JFM 49.0007.02)] proved that for every even \(n\) there exists a latin square of order \(n\) that has no orthogonal mate, and in 1944 H.B. Mann [“On orthogonal Latin squares”, Bull. Am. Math. Soc. 50, 249–257 (1944; Zbl 0060.32307)] proved that for every \(n\) of the form \(4k+1\), \(k \geq 1\), there exists a latin square of order \(n\) that has no orthogonal mate. Except for the two smallest cases, \(n = 3\) and \(n = 7\), it is not known whether a latin square of order \(n = 4k+3\) with no orthogonal mate exists or not. We complete the determination of all \(n\) for which there exists a mate-less latin square of order \(n\) by proving that, with the exception of \(n = 3\), for all \(n = 4k+3\) there exists a latin square of order \(n\) with no orthogonal mate. We will also show how the methods used in this paper can be applied more generally by deriving several earlier non-orthogonality results.
[Author’s Note: The main result of this interesting paper was also independently established by I.M. Wanless and B.S. Webb [Des. Codes Cryptogr. 40, No.1, 131–135 (2006; Zbl 1180.05023)].