Dan, you are right, there is a great relationship between additive and multiplicative squares (and cubes), but not always a direct relationship. For example the beauty of the multiplicative squares and cubes is to find the smallest possible entries and the smallest possible products, and -as far as I know- there is no direct way to find this, using additive squares. That's the open problems of my 2 tables at www.multimagie.com/English/Multiplicative.htm and www.multimagie.com/English/MultiplicCubes.htm where the max nbs and products are the "smallest known". Only few of them are proved to be the smallest possible. I am sure that numerous results of these tables can be improved. Another problem: pandiagonal additive squares (normal magic square, using consecutive integers) of orders 4k+2 are impossible, as first proved by Frost in 1878. But we know now that 6x6 and 10x10 pandiagonal multiplicative square are possible. Because normal pandiag. squares of such orders cannot exist, for sure we can't use them for pandiag. multiplicative squares! Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de dasimov@earthlink.net Envoyé : dimanche 23 avril 2006 16:48 À : math-fun Objet : Re: [math-fun] 6x6 pandiag. multiplic. squares ARE NOT impossible! Michael Kleber's latest brilliant analysis of Christian's new multiplicative magic square -- in terms of bit planes -- is quite reminiscent of (what I think it) Conway & Coxeter's analysis of additive magic squares as it appears in (some of) the most recent edition(s) of W.W. Rouse Ball's "Mathematical Recreations and Essays" (updated by Coxeter). Let N := the set of nonnegative integers. Then {additive magic squares of elements of N^k for some k} is in 1-1 correspondence with {multiplicative magic squares} -- just letting the components of v in N^k correspond to prime exponents. I was wondering if there could be a multiplicative magic square that was somehow not a consequence of additive ones . . . but now I see that's impossible. --Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun