Start with the old 4x4 magic square construction 1 - - 4 - 15 14 - - 6 7 - merged with 12 - - 9 - 10 11 - 8 - - 5 13 - - 16 - 3 2 - Subtract 1 (to make the range 0-15) and then take binary bit planes. This gives the patterns --xx -x-x -xx- -xx- xx-- x-x- -xx- x--x xx-- x-x- x--x -xx- --xx -x-x x--x x--x and each row, column, and diagonal has two xs. Stacking the four patterns gives all 16 possible Z combinations. Assign the primes 2,3,5,7 to the four patterns. Multiply the pattern entries, getting all the divisors of 210 as the square entries. The magic product is 210^2 which is 44100. We can lower the product a little by using 4 in place of 7; the product is then 14400. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com]On Behalf Of ed pegg Sent: Wednesday, September 21, 2005 3:50 PM To: math-fun Subject: RE: [math-fun] Multiplicative Magic Squares If you just use powers of 2, the minimal square with distinct entries has multiplicative constant 1073741824 (2^30). 55440 is much smaller. Is it the smallest? --Ed Pegg Jr --- "Torgerson, Mark D" <mdtorge@sandia.gov> wrote:

Replacement goes a long way. You may take all the 3s in the square below and turn them into 29s. Or some other square having large primes may be replaced with a square with smaller corresponding primes. Even primes within a particular square may be switched. 55440=2^4*3^2*5*7*11 so you can make a corresponding square with 2*3^2*5*7*11^4. No matter the starting square, this reduction leads to some sort of minimal representation, where the factorization of the product gives 2 with the largest exponent, 3 the next and so on.

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