A construction for doubly pandiagonal magic squares (Q658081)

The authors introduce a product construction for doubly magic rectangles and a construction for doubly pandiagonal magic squares. A \textit{magic square} is a square array containing consecutive integers (usually beginning with \(0\) or \(1\)) in which every row, column, and diagonal adds to the same ``magic'' constant. Further, a magic square is \textit{pandiagonal} if all the broken diagonals also add to the magic constant. People have been amusing themselves with magic squares for thousands of years [\textit{W.\ W.\ Rouse Ball} and \textit{H.\ S.\ M.\ Coxeter}, Mathematical Recreations and Essays, Dover, (1987)] and they have become useful (along with latin squares) in statistical design [\textit{J.\ Dénes} and \textit{A.\ D.\ Keedwell}, Latin Squares and their Applications. Budapest: Akademiai Kiado (1974; Zbl 0283.05014)]. A magic square is said to be \textit{doubly magic} if when one squares each entry the resulting array has rows, columns, and diagonals that add to a (new) magic constant sum; it is called \textit{doubly pandiagonal magic} if both it and its squared counterpart are pandiagonal. The first doubly magic squares were given by \textit{G.\ Pfefferman} [``Les tablettes du chercheur'', J. des Jeux d'Espririt et de Combinaisons, Paris (1891)] and are discussed also in [\textit{H.\ Derksen, C.\ Eggermont} and \textit{A.\ van den Essen}, ``Multimagic Squares'', Am. Math. Mon. 114, No. 8, 703--713 (2007; Zbl 1141.05024)]. A \textit{magic rectangle} is a non-square array of consecutive integers in which each row adds to a constant sum and each column adds to a (different) constant sum; \textit{doubly magic rectangles} are defined similarly to doubly magic squares. The main results of the article are: {\parindent=6mm \begin{itemize}\item[(1)]Given doubly magic rectangles of size \(p\times q\) and \(u\times v\), there exists a doubly magic rectangle of size \(pu\times qv\). \item[(2)]If there exists a \(p\times q\) doubly magic rectangle (here \((p,q)=1\)) then there exists a doubly pandiagonal magic square of order \(pq\). \item[(3)]For each \((p,q)\in \{ (11,7), (13,7), (19,7), (13,11), (17,11)\}\) there exists a doubly pandiagonal magic square of order \((pq)^n\) for each \(n\in {\mathbb Z} ^+\). \end{itemize}} If \(A=(a_{ij})\) is doubly magic of order \(pq\) (\(p,q\) relatively prime) then one can define a bijection \(f:{\mathbb Z}_{pq}\rightarrow {\mathbb Z}_{pq}\) by \(f(\langle qi+pj \rangle _{pq})=a_{ij}\). The doubly pandiagonal magic square from item (2) above has entries \[ h_{x,y}=pqf(\langle px+(q-p)y\rangle _{pq})+ f(\langle (p+q)x+py\rangle _{pq}) \] for \(x,y\in {\mathbb Z}_{pq}\).