Here is the current status of small add-mult squares: www.multimagie.com/English/SmallestAddMult.htm Today, no known example smaller than 8x8. I also think that 5x5 (and 6x6 and 7x7) should be possible, a very interesting problem! You will see the proof that 4x4 add-mult magic are impossible, but you are right, dropping the diagonals = 4x4 "semi"-magic are possible. Here is one of the examples found by Lee Morgenstern, and given in the above webpage: 110 72 63 80 64 105 66 90 81 88 100 56 70 60 96 99 Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de Schroeppel, Richard Envoyé : dimanche 1 février 2009 04:50 À : math-fun@mailman.xmission.com Objet : [math-fun] add-mult magic squares I'm curious about small addition-multiplication squares. Assume we ditch the integer requirement, and allow real or complex numbers - presumably algebraic. We keep the requirement that the elements are distinct. If we drop the diagonals, then simple counting of constraints suggests there should be a 4x4 with a couple of degrees of freedom remaining; and there should be 5x5s for which the diagonals work. This seems like a good problem for multiple approaches: multivariate hill climbing; or some heavy-duty algebra. Rich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun