List of the site's news added August 16th, 2013

• Death on July 6th 2013 of Harvey Heinz (biography), Canada, the author of www.magic-squares.net.
  He checked the English of my website for more than ten years, it was a pleasure and an honor to work with him.
          Harvey in 2003

• New best known multiplicative magic squares or cubes
• New best known 6x6 pandiagonal multiplicative magic square, by Radko Nachev, and updated table of best known pandiagonal multiplicative magic squares

1	36	600	80	120	225
300	1800	3	5	144	40
200	4	360	150	15	72
45	30	16	3600	100	6
720	25	90	12	2	1200
24	240	50	18	900	10

• New best known multiplicative magic squares (8x8 and 9x9) by Toshihiro Shirakawa, and updated table and file of best known multiplicative magic squares
• New best known add-mult magic squares (orders 12, 13, 14, 15, 17, 19, 21) by Toshihiro Shirakawa, and updated table and file of best known add-mult magic squares
           Even if their additive properties could be a handicap, all these squares are also the new best known multiplicative magic squares!
• New best known multiplicative cubes by André LFS Bacci (semi-magic 3x3x3) and Toshihiro Shirakawa (perfect magic 5x5x5 and 6x6x6), with updated table and file

• NEW RECORDS! New smallest known orders of semi-magic or of magic squares of powers!
• Two 64x64 magic squares of 6th powers, first one from Jaroslaw Wroblewski, second one later from Toshihiro Shirakawa, which are smaller than Labelle's 90x90 (the previous record)
• 64x64 semi-magic square of 7th powers by Toshihiro Shirakawa, smaller than his 81x81, and than the 196x196 by Jaroslaw Wroblewski (the previous records)
• With these new results, here is the updated table of smallest known squares of powers:

Lucas	Smallest known squares	semi-magic	magic	Euler
	of squares	3x3, Edouard Lucas (1876)	4x4 Leonhard Euler (1770)
	of cubes	4x4, Lee Morgenstern (2006)	8x8, Walter Trump (2008)
	of 4th powers	4x4, Morgenstern-Boyer (2006)	15x15, Labelle-Boyer (2011)
	of 5th powers	16x16, J. Wroblewski (2011)	36x36, Li Wen (2008)
	of 6th powers	36x36, J. Wroblewski (2011)	64x64, J. Wroblewski (2012) NEW!
	of 7th powers	64x64, T. Shirakawa (2013) NEW!	65536 x 65536

• But smallest possible squares not yet known! Who will find smaller squares?
           Only 3x3 semi-magic squares of squares are proved impossible to improve (2x2 being of course impossible...).
           The goal of several enigmas (#1, #3, #4, #4a, #4b, #4c) is to improve this table.
• Bigger than 15x15 by Labelle-Boyer, but an interesting new 16x16 magic square of 4th powers, and its new construction method, by Toshihiro Shirakawa

• There still remain ten enigmas for winning €6,900 and ten bottles of champagne!
  Here are the most interesting results received on the remaining enigmas, but still unsolved:
• #1 (3x3 magic square of squares, or using at least 7 distinct squared integers), interesting contributions from Mike Winkler, Lee Morgenstern, Tim S. Roberts
• #3 (3x3 semi-magic square of cubes) no solution with cubes < (10^6)^3 by Lee Morgenstern
• #4 (4x4 magic square of cubes)
• no solution using taxicab numbers < 9*10^18, giving magic sums up to 8.1*10^37, by Gildas Guillemot
• no solution with magic sum < 2*10^18, but 4x4 magic square of 14 cubes and new search method, by Lee Morgenstern
• #4c (7x7 magic square of cubes) no solution with magic sum < 418000, but six semi-magic squares with one magic diagonal, by Toshihiro Shirakawa

• New results on multimagic series
• Formulas estimating the number of bimagic and trimagic series for squares, and formula estimating the number of bimagic series for cubes, by Michael Quist
           Quist's formula, number of bimagic series for squares of order N
• Numbers of bimagic series for squares of orders from 21 to 28 computed by Michael Quist, Lee Morgenstern, and Walter Trump.
           The number of bimagic series of order 28 is as large as 2455456581123976779162274548131606029110869.
• Numbers of bimagic series for cubes of orders 11 and 12 computed by Walter Trump using a program written by Lee Morgenstern
• Impossibility of some multimagic series for squares proved by Lee Morgenstern
           No tetramagic series of order 17. No pentamagic series of orders 19 and 23. No hexamagic series of order < 40, except hexamagic 27 which remains unknown.

• New published papers
• in the Journal of Combinatorial Designs, multimagic theorem by Yong Zhang, Kejun Chen and Jianguo Lei (improving results of Harm Derksen, Christian Eggermont and Arno van den Essen)

n-multimagic squares of order pn exist for any prime p ≥ 2n - 1 with n ≥ 2

• in Discrete Mathematics, construction method of bimagic squares, by Kejun Chen and Wen Li
• in Mathematics Today, pandiagonal sudokus by Ronald P. Nordgren

• And also...
• Tetramagic squares of orders 20 to 23, and 25, are impossible, by Lee Morgenstern. Maybe the smallest possible tetramagic square is of order 24?
• 4x4 nearly bimagic squares, with 19 bimagic lines out of 20, are impossible, by Lee Morgenstern
• The smallest possible order of axially symmetric bimagic squares is 12, by Francis Gaspalou and Walter Trump
• "Centralsymmetric" perfect magic cubes 5x5x5 are impossible, by H. B. Meyer
• Magic squares of hexagonal to decagonal numbers, by Christian Boyer and by Lee Morgenstern, and updated table of smallest squares of polygonal numbers
• Magic squares of primes in arithmetic progression
  and not in this website, but see Al Zimmermann's Programming Contest on pandiagonal magic squares of primes, orders 6 to 20, with impressive results from Jaroslaw Wroblewski in August 2013
• Modified biography of A.H. Frost, with new interesting info from John Williamson:
  Frost's cube of eleven, and Firth's cube of six dedicated to Frost, found at Royal Holloway University of London. There remains a last lost cube: who will find Frost's cube of seven?
            Frost's cube of eleven

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