There's an interesting oblique way to look at these. If a square is mul(tiplicatively ma)gic, then taking logs gives you a regular magic square (albeit with transcendental entries). Equally, looking at the exponents of particular primes also gives a magic square. So the log-square falls in the class of squares that are (additively) magic, and also magic if a function F() is applied to the cells. Simpler examples have F() = squaring; there may be examples with both squaring & cubing. --- The 3x3 magic squares are parameterized as K+a K-a-b K+b K-a+b K K+a-b K-b K+a+b K-a I don't know of a pretty parameterization for the 4x4s. You can pretty much pick any 7 cells (with no lines or other forced totals), and the sum S. a b c - d e f - g - - - - - - - Just fill in the other values to make the total right. Remember the sum of the 4 corners is also S. 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 David Wilson Sent: Friday, September 23, 2005 8:33 AM To: Michael Kleber; math-fun Subject: Re: [math-fun] Multiplicative Magic Squares Nice argument. Can we satisfy a+b+c = abc two different ways as a first step towards a 4x4? ----- Original Message ----- From: "Michael Kleber" <michael.kleber@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Friday, September 23, 2005 10:02 AM Subject: Re: [math-fun] Multiplicative Magic Squares David Wilson wrote:
Can a 3x3 magic square be both additively and multiplicatively magic?
Not even close, even if you don't try for diagonals. There's at most one solution to {x+y=S, xy=P} for any S and P -- as we were taught when we learned to factor polynomials -- and any z in a both-ways-magic square would force its row-mates and column-mates to be two solutions to such a system. Nice work on the possible 4x4 magic products, by the way... --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun