Variations

If you have constructed one composite magic square, a great variety of other, non-equivalent magic squares can be derived. Let us take a look at the following magic square.

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Each of the nine subsquares is magic and can be rearranged with one of the usual rotations or reflections without losing the magic quality. But this means that also the magic quality of the fundamental magic square remains.

If we rotate e.g. the centered subsquare by 90° and leave the other subsquares unchanged, we have created another non-equivalent magic square (left fig.). We also may reflect the centered subsquare at his main diagonal (right fig.).

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646872272319363228
697065222126313035
834384540747978
159434139817773
672423744767580
475449566358111615
525048615957181410
514653605562131217
716667202524293433
646872272319363228
697065222126313035
834443940747978
159374145817773
672424338767580
475449566358111615
525048615957181410
514653605562131217

You also may rearrange two, three or even all subsquares at the same time without affecting the magic quality of the fundamental square. These ideas show that hundreds and thousands of non-equivalent magic squares can be derived from one existing composite magic square.