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A Magic Square of the Order 4

One way to construct a magic square of the order 4 (16 cells using the numbers 1 to 16) is shown below.

(1) Put in the numbers 1 to 16 starting from the top left-hand corner.

(2) Swap numbers in the opposite diagonal corners...

(3) .. now swap the numbers in the middle 4 cells.

Try making a magic square with this starting arrangement:

The numbers 1 to 16 were placed in the square starting in the top left-hand corner and going down the column.

There are 880 different arrangements, not reflections or rotations, for the numbers 1 to 16 in a magic square of the order 4. How many can you find using this method?