Nice square by hand, Ed! By program, it is possible to construct other semi-magic 4x4 squares like yours with magic product 4320. But IF my program (written perhaps too quickly...) is correct, it is impossible to get at least one diagonal = 4320, when all rows and columns = 4320. The nearest possible diagonal = 4608, for example with: 16 1 10 27 5 24 18 2 6 12 3 20 9 15 8 4 It means that the smallest possible magic product for 4x4 seems to be 5040. 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 ed pegg Envoyé : vendredi 23 septembre 2005 06:24 À : math-fun Objet : Re: [math-fun] Multiplicative Magic Squares
So the smallest magic product of a 4x4 square of positive integers is either 4320 or 5040. Presumably someone can check whether 4320 is a magic product.
Nice analysis! By hand (the programming was getting nowhere), I found 1 27 8 20 10 16 9 3 24 2 15 6 18 5 4 12 Which has a constant of 4320 on the rows and columns, but not the diagonals. With further noodling, perhaps the diagonals can be fixed. I'm fairly certain Dudeney's 60,466,176 for the 5x5 can be beaten. Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun