Geodesic
On melancholic magic squares
Discussiones Mathematicae. Probability and Statistics, Tome 33 (2013) no. 1-2, pp. 111-119

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Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares.
Keywords: magic squares, patterns, group inverse, Moore-Penrose inverse, eigenvalues and eigenvectors 
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Trenkler, Götz; Trenkler, Dietrich. On melancholic magic squares. Discussiones Mathematicae. Probability and Statistics, Tome 33 (2013) no. 1-2, pp. 111-119. http://geodesic.mathdoc.fr/item/DMPS_2013_33_1-2_a7/