The magic square legend

According to a legend , there was at one time in ancient China a huge flood. While the great king Yu was trying to channel the water out to sea, a turtle emerged from it with a curious pattern on its shell (see the image below): a magic square, i.e. a n x n nxn square grid filled with distinct positive integers in the range 1 , 2 , . . . , n 2 1,2,...,n^2 such that each cell contains a different integer and the sum x x of the integers in each row, column and diagonal is equal. Let the magic constant be 30 30 , how many different square grids with order 3 3 exists?

Assumptions : changing a row determines two different magic squares (e.g. changing the second row 9 + 5 + 1 9+5+1 with 2 + 11 + 3 2+11+3 ) as well as invert the order of numbers (e.g. 3 + 4 + 8 3+4+8 and 4 + 3 + 8 4+3+8 ); the same for columns. Furthermore there is not a limit on the maximum number inside each squares of the grid.


The answer is 104.

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