The Number of "Magic" Squares, Cubes, and Hypercubes

Magic squares have turned up throughout history, some in a mathematical context, others in philosophical or religious contexts. According to legend, the first magic square was discovered in China by an unknown mathematician sometime before the first century A.D. It was a magic square of order three thought to have appeared on the back of a turtle emerging from a river. Other magic squares surfaced at various places around the world in the centuries following their discovery. Some of the more interesting examples were recorded in Europe during the 1500s. Cornelius Agrippa wrote De Occulta Philosophia in 1510. In it he describes the spiritual powers of magic squares and produces some squares of orders from three up to nine. His work, although influential in the mathematical community, enjoyed only brief success, for the counter-reformation and the witch hunts of the Inquisition began soon thereafter: Agrippa himself was accused of being allied with the devil. Although this story seems outlandish now, we cannot ignore the strange mystical ties magic squares seem to have with the world and nature surrounding us, above and beyond their mathematical significance. Despite the fact that magic squares have been studied for a long time, they are still the subject of research projects. These include both mathematical-historical research, such as the discovery of unpublished magic squares of Benjamin Franklin [12], and pure mathematical research, much of which is connected with the algebraic and combinatorial geometry of polyhedra (see, for example, [1], [4], and [11]). Aside from mathematical research, magic squares naturally continue to be an excellent source of topics for “popular” mathematics books (see, for example, [3] or [13]). In this paper we explore counting functions that are associated with magic squares. We define a semi-magic square to be a square matrix whose entries are nonnegative integers and whose rows and columns (called lines in this setting) sum to the same number. A magic square is a semi-magic square whose main diagonals also add up to the line sum. A symmetric magic square is a magic square that is a symmetric matrix. A pandiagonal magic square is a semi-magic square whose diagonals parallel to the main diagonal from the upper left to the lower right, wrapped around (i.e., continued to a duplicate square placed to the left or right of the given one), add up to the line sum. Figure 1 illustrates our various definitions. We caution the reader about clashing definitions in the literature. For example, some people would reserve the term “magic square” for what we will call a traditional magic square, a magic square of order n whose entries are the integers 1, 2, . . . , n2. Our goal is to count these various types of squares. In the traditional case, this is in some sense not very interesting: for each order there is a fixed number of traditional magic squares. For