Excellent and very recent news: first known 6x6 bimagic square using distinct integers found by Jaroslaw Wroblewski, University of Wroclaw, Poland. 3x3 and 4x4 were previously proved impossible, and 6x6 is now proved possible. But the existence of a 5x5 bimagic square is still unknown! Question: do you know if it is possible to submit the corresponding set of 24 equations to a mathematical package? If somebody can submit the problem to his math product, could you report me the results? Perhaps that even a solution with 25 distinct real numbers (instead of distinct integers) is impossible? If the 5x5 square is a b c d e f g h i j k l m n o p q r s t u v w x y and if the magic sums are S1 and S2, then the set of equations is: a + b + c + d + e = S1 a² + b² + c² + d² + e² = S2 f + g + h + i + j = S1 f² + g² + h² + i² + j² = S2 ... 24 equations (2*5 for rows + 2*5 for columns + 2*2 for diagonals). And the 25 used numbers, from a to y, have to be distinct. Today, the nearest known solution is a 5x5 square solving a subset of 22 equations on 24: 3 37 20 44 16 34 35 1 12 38 41 8 24 40 7 10 36 47 13 14 32 4 28 11 45 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 : mercredi 19 octobre 2005 11:02 À : 'math-fun' Objet : [math-fun] Smallest bimagic square: unknown! A magic square is bimagic if it remains magic after squaring each of its integers. The smallest bimagic squares using consecutive integers are known: 8x8. BUT... the smallest bimagic squares using distinct integers (= not forced to be consecutive) are still unknown! 5x5, 6x6, 7x7??? If 5x5 is impossible, a proof??? Edouard Lucas was the first to work on the subject, in 1891, easily proving that 3x3 is impossible. For more information, go to www.multimagie.com/indexengl.htm Click on "Magic squares of squares" in the left menu, then click on "Open problems". And, from this table, you will find the current status of the research. Any help or idea on the subject is welcome! Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun