The Number of Solutions to a Magic Square

A magic square is a 3 by 3 square so that the numbers 1 to 9 each appear once, and the numbers in each row, column or diagonal add up to the same number.

There are three rows and three columns so each rown, and each column must add up to 15.
There are only eight possible sums of three numbers to give 15.
1 + 5 + 9 = 15
1 + 6 + 8 = 15
2 + 4 + 9 = 15
2 + 5 + 8 = 15
2 + 6 + 7 = 15
3 + 4 + 8 = 15
3 + 5 + 7 = 15
4 + 5 + 6 = 15
5 appears four times, so must be at the centre of the square, as the sum of the middle row, the middle column and two diagonals
2, 4, 6, 8 each appear three times so must be at the corners, to appear in the sum of a row, diagonal and columns. The square can then only be completed in one way, giving one solution.
Hence, there is at least one solution, namely

Other solutions can be obtained from this one using the symmetries of a square. The group of symmetries of a square consists of four rotations and four reflections - one horizontal, one vertical and two diagonal.
There are eight solutions in total.