How many *k* × *k* magic squares are possible? If you start from a liberal definition of magic square, there’s an elegant result. For the purposes of this post, a magic square is a square arrangement of non-negative numbers such that the rows and columns all sum to the same non-negative number *m *called the *magic constant*. Note that this allows the possibility that numbers will be repeated, and this places no restriction on the diagonals.

With this admittedly non-standard definition, the number of *k* × *k* magic squares with magic constant *m* is always a polynomial in *m* of degree no more than *(k* – 1)^{2}. For *k* = 3, the result is

(*m* + 1)(*m* + 2)(*m*^{2} + 3*m* + 4)/8

There is no general formula for all *k*, but there is an algorithm for finding a formula for each value of *k*.

Source: The Concrete Tetrahedron

**Update**: I had reported the polynomial degree as *k* + 1, but looking back at Concrete Tetrahedron, that book lists the order as (*k* + 1)^{2}. However, the paper cited in the comments lists the exponent as (*k* – 1)^{2}, which I believe is correct.

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