Dear friends, It seems impossible to get better 6x6 multiplicative squares that my examples found two days ago. So: the smallest possible magic product for 6x6 multiplicative squares should be 25945920 = 2^6 x 3^4 x 5 x 7 x 11 x 13. Here is one of the numerous examples: 1 22 39 54 28 20 12 65 18 8 3 77 30 27 14 52 11 4 42 16 6 33 15 13 26 21 44 5 24 9 66 2 10 7 78 36 This example uses the smallest possible set of integers (1..78) for that magic product. But is interesting to note that it is possible to construct 6x6 multiplicative squares with a more compact set of integers (1..66), but with a... bigger magic product! Here is an example with magic product 39916800: 8 45 42 66 1 40 50 56 44 3 54 2 16 33 5 28 30 18 21 10 15 9 64 22 27 4 6 32 55 35 11 12 48 25 7 36 The 10 smallest magic products of 6x6 multiplication squares should be: 25945920 26611200 28828800 29937600 31449600 33264000 33929280 34594560 34927200 35380800 I have the 40 smallest 6x6 magic products that I will propose to Neil for his OEIS. Christian. -----Message d'origine----- De : Christian Boyer [mailto:cboyer@club-internet.fr] Envoyé : vendredi 14 octobre 2005 08:42 À : 'math-fun' Objet : RE: [math-fun] 6x6 multiplicative magic squares Latest (and good) news! I can construct a lot of 6x6 multiplicative squares. My current smallest 6x6 multiplicative magic square has a magic product more than 70 times smaller than the Borkovitz & Hwang square. The B&H square used 36 distinct integers, the biggest being 1,260. My current best 6x6 example uses also 36 integers (of course...), but the biggest integer is 78: an extremely compact set of integers! But I need more checking before to be sure that it will be impossible to get better 6x6 results, and will keep you informed. 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 Christian Boyer Envoyé : lundi 10 octobre 2005 12:07 À : 'math-fun' Objet : [math-fun] 6x6 multiplicative magic squares After 3x3, 4x4, 5x5, what about 6x6? My algorithm needs to be better optimized to attack the question. I will work on it in the next days. The smallest published 6x6 seems to be an example constructed by Borkovitz & Hwang, with magic product = 2,000,376,000. My question: can somebody produce an example with a smaller product? Michael Kleber? Rich? Unfortunately the latin squares method, used in previous messages for 4x4 and 5x5, does not work for 6x6: the famous "36-officers problem" of Euler has no solution! Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun