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Problem of the Week 1056

Multiplicative Magic Square

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Find a 3 × 3 array of distinct numbers so that the product of every row, of every column, and of each of 4 of the 6 diagonals is the same.

Note. "Numbers" here means real or complex numbers and "diagonal" means "generalized diagonal": any triple having one entry in each row and one in each column. There are six of these corresponding to the permutations of {1, 2, 3}.

Extra Credit. Determine, for the MMC (multiplicative magic square) that you find, how many magic transposals it has. By this we mean: How many ways are there of permuting the numbers into an essentially different MMC, where we count two MMC as different if they are not rotations or reflections of each other? Note that the definition of an MMC is that the product of every row, every column, and the two main diagonals are the same.

Source: Lee Sallows, Nijmegen, The Netherlands.

© Copyright 2006 Stan Wagon. Reproduced with permission.

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5 April 2006