Polyhedral Cones of Magic Cubes and Squares

Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes. Magic cubes and squares are very popular combinatorial objects (see [2, 15, 17] and their references). A magic square is a square matrix whose entries are nonnegative integers and whose row sums, column sums, and main diagonal sums add up to the same integer number s. We will call s the magic sum of the square. In the literature there have been many variations on the definition of magic squares. For example, one popular variation of our definition adds the restriction of using the integers 1, . . . , n as entries (such magic squares are commonly called natural or pure and a large part of the literature consists of procedures for constructing such examples, see [2, 15, 17]), but in this article the entries of the squares will be arbitrary nonnegative integers. We will consider other kinds of restrictions instead: Semi-magic squares is the case when only the row and column sums are considered. This apparent simplification has in fact a very rich theory and several open questions remain [7, 22]. Pandiagonal magic squares are magic squares with the additional property that any broken-line diagonal sum adds up to the same integer (see Figure 1).