A magic squareM is an n × n matrix with entries 1, . . . , n~2 such that the sums over each of the n rows, each of the n columns, and each of the two diagonals, have the same value. There are many different subclasses of magic squares, and the construction, and enumeration, of all types of magic squares presents a substantial challenge, largely because of the lack of an algebraic structure on the set {1, . . . , n~2}. A mathematical treatment of magic squares is more fruitful if we allow their entries to be real numbers, and then consider them to be elements of a vector space of real matrices, in which case we shall use the term magic matrix. For us, then, a magic matrix is a real n × n matrix M such that the sums over each of its n rows, each of its n columns, and each of its two diagonals, have the same value. We emphasize that, in contrast to magic squares, the elements of a magic matrix need not be integers, nor distinct. The word “magic” was first introduced in this context by Frenicle de Bessy [6] in 1629; more recently the term “magic matrix” rather than “magic square” was used in [4, 5].
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