Orthogonal Latin magic squares: Some applications and some open problems (Q2731009)

The author defines a Latin magic square as a Latin square with distinct entries on each diagonal. By this we assume the author means a Latin square \(A = (a_{i,j})\) of order \(n\) with distinct entries in cells \(a_{i,i}\), \(1 \leq i \leq n\) and distinct entries in cells \(a_{i,n+1-i}\), \(1 \leq i \leq n\). The paper poses a series of problems. The first two ask for a pair of mutually orthogonal Latin magic squares of all orders greater than 3. This problem was in fact solved some time ago by \textit{K. Heinrich} and \textit{A. J. W. Hilton} [Discrete Math. 46, 173-182 (1983; Zbl 0517.05013)] for all but six values of \(n\), five of which have since been settled, leaving only the case \(n = 10\). Other problems include: How many magic square can be constrcuted with arbitrary first row (magic squares here are not required to have sequential entries---just the row, column and diagonal sums must be constant)? How many magic squares are there with the additional property that the sum of the digits of the entries in each row, column and diagonal is constant?