Magic Square

A Magic Square is a type of puzzle in recreational mathematics where a square grid of any dimension is filled with numbers such that each row, column and diagonal has the same sum. The length of the side is called its order and the target sum is called the magic constant. In the example below, the order is 3 and the magic constant is 15.

When placing numbers into the grid, you must place positive integers starting with 1 and increasing to the area of the grid. In the example above, the area of the grid is 9 so the numbers that must be placed are 1-9. In a magic square of order 4 you would place the numbers 1-16.

There is exactly one magic square of order 1. There are no magic squares of order 2. For order 3, there is only one magic square (see above) when you eliminate rotations and mirror images. For order 4, there are 880 solutions and for order 5 there are more than 275 million solutions. How many can you find?