A semimagic square is a square that fails to be a magic square only because one or both of the main diagonal sums do not equal the magic constant. Note some care with terminology is necessary. For example, Jelliss terms a semimagic square a "magic square" and a magic square a "diagonally magic square." The number of distinct semimagic squares (treating squares differing by rotations and reflections as identical) of orders 1, 2, ... are 1, 0, 8, .... The eight semimagic squares of order 3 are illustrated above.
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