Enigmas on Magic Squares: win €8,000 and 12 bottles of champagne!!!

 While magic squares have been known and studied for many centuries, it is surprising that for certain types of magic squares we still do not know today which are the smallest possible! In an effort to make progress on these unsolved problems, twelve prizes totaling €8,000 and 12 bottles of champagne are offered for the solutions to twelve enigmas (six main at €1,000 each, six small from €100 to €500 each): read the press release of April 2010... and send me your solutions! With the solutions of enigmas #3a, #4c, #5 and #6b, there still remain eight prizes totaling €6,500 + eight bottles of champagne (at the time of the last update of this website). Since all the enigmas on 7x7 squares are now solved, the remaining enigmas are on small squares, from 3x3 to 6x6.

Who can construct, or prove the impossibility:

1. 3x3 magic square using 7 (or why not 8, or 9) distinct squared integers different from this only known example (and of its rotations, symmetries and k² multiples):
1.  373² 289² 565² 360721 425² 23² 205² 527² 222121
2. 5x5 bimagic square using distinct positive integers
3. 3x3 semi-magic square of cubes using distinct positive cubed integers (small enigma #3a: square 7x7)
4. 4x4 magic square of cubes using distinct positive cubed integers (small enigmas #4a, #4b, #4c: squares 5x56x6, 7x7)
5. multiplicative magic cube using distinct positive integers < 364
6. 5x5 additive-multiplicative magic square using distinct positive integers (small enigmas #6a , #6b: squares 6x67x7)

No, I myself do not have the solutions... Of course, only the first person who solves an enigma will win the associated prize and will be named in this table:

 Magic squares of squares Bimagic squares Semi-magic squares of cubes Magic squares of cubes Add-mult magic squares 2x2 Impossible 3x3 Main enigma #1(€1000)* Impossible. Proved by E. Lucas, 1891 Main enigma #3(€1000) Impossible Impossible. Proved by L. Morgenstern, 2007 4x4 L. Euler, 1770 Impossible. Proved by L. Pebody / J.-C. Rosa, 2004** L. Morgenstern, 2006 Main enigma #4(€1000) 5x5 C. Boyer, 2004 Main enigma #2(€1000) C. Boyer, 2004 Small enigma #4a(€500) Main enigma #6(€1000) 6x6 C. Boyer, 2005 J. Wroblewski, 2006 L. Morgenstern, 2006 Small enigma #4b(€500) Small enigma #6a(€500) 7x7 C. Boyer***, 2005 L. Morgenstern, 2006 Small enigma #3aT. Shirakawa, 2010 Small enigma #4cS. Miquel, 2015 Small enigma #6bS. Miquel, 2016 8x8 G. Pfeffermann***, 1890 L. Morgenstern, 2006 W. Trump, 2008 W. Horner, 1955 9x9 G. Pfeffermann***, 1891 L. Morgenstern -C. Boyer, 2006 C. Boyer***, 2006 W. Horner, 1952

* or using at least 7 squared integers among its 9 integers, different from the only known example
** proved the same year, but independently
*** these squares use consecutive integers (or consecutive squared integers, or consecutive cubed integers)
Countries: Switzerland (Euler), England (Pebody), France (Pfeffermann, Lucas, Rosa, Boyer, Miquel), Germany (Trump), Japan (Shirakawa), Poland (Wroblewski), USA (Horner, Morgenstern)

Important remark on the main enigma #1. Strictly speaking, an impossibility proof of 8 or 9 distinct squared integers in a 3x3 magic square is not a solution, because another 3x3 magic square using 7 squared integers remains (perhaps) possible. However, because such an impossibility proof would be an impressive result, it will be rewarded by a prize: €500 + bottle of champagne.

Winners

 Congratulations to Toshihiro Shirakawa, Japan, who, very quickly after the announcement of the contest on April 6th, 2010, solved two enigmas:#5 as soon as April 15th with his cube, then #3a one week later, April 22nd, with his square Here happy, with his first bottle of champagne! He received a second one, some days later. A total of two Moët & Chandon impérial bottles, and of €1100. Congratulations to Sébastien Miquel, France, who solved the enigma #4c, February 20th, 2015,with his square and so won a bottle and €200. C. Boyer & S. Miquel(Paris, March 2015) A year and a half later, congratulations again to Sébastien Miquel, who solved the enigma #6b, August 15th, 2016,with his square and so won again a bottle and €200. Who will be the next winner? With which enigma? C. Boyer & S. Miquel(Paris, September 2016)

Enigmas in Pour La Science... and elsewhere

in
Dossier Pour La Science (Jeux math')....... and Pour La Science website

• In "Enigmes sur les Carrés Magiques", paper published in the Dossier Pour La Science of April-June 2008 (N°59, pages 22-25), I offered €100 + a bottle of champagne for each of the first 5 main enigmas.
• One year later in the Pour La Science website, the same 5 main enigmas were republished http://www.pourlascience.fr/ewb_pages/j/jeux.php (April-May 2009) and a 6th new enigma was added (June 2009).
• Two years later, April 2010: €1,000 are now on offer for each of these six main enigmas, and six small enigmas from €100 to €500 are added.

Many thanks to the numerous people, magazines and websites for announcing the contest after receiving the press release sent April 6th 2010, in particular, in chronological order:

In advance, sorry to others of whom I am not aware, but I also thank them!

And also, for reporting the solutions found by Toshihiro Shirakawa, thanks to:

• Akira Iino, Sugaku Seminar, Vol 49, N°12 591, 2010-12, p.38-41 (long paper of 4 pages written in Japanese, also reporting in details the 12 enigmas)
• Michel Criton, Tangente, N°138, January-February 2011, p.6

With this Japanese paper in Sugaku Seminar, I know now that my name is written that way in katakana:

Christian (underlined in red) Boyer (underlined in blue) written in katakana!

Thanks to Toshihiro for identifying the characters of my name. Amusing: the numbers being the only characters that we can easily read, we may deduct that this paragraph probably says that in 2010, in April (4), I submitted 12 enigmas, prizes totaling 8000 euros, each being from 100 to 1000 euros. Am I right?

Thanks for their report of the solution of #4c found by Sébastien Miquel:

• , Ivars Peterson, MathTourist Twitter, March 18, 2015
• David Larousserie, Le Monde, March 25, 2015, Science & Médecine p.3, downloadable PDF
• Philippe Fondanaiche, Diophante.fr, April 2015
• La Vie de l'Ecole (Ecole Normale Supérieure, rue d'Ulm, Paris), N°3, April 2015, p.1, downloadable PDF
•   Pour La Science, N°451, May 2015, p.10
• Edouard Thomas, Tangente, N°164, May-June 2015, p.2
• Matt Parker and Brady Haran, Numberphile, The Parker Square, April 2016, YouTube video (1:02-1:36), also on enigma #1

Thanks for their report of the solution of #6b found by Sébastien Miquel:

• Ivars Peterson, MathTourist Twitter, February 18, 2017
•  Philippe Ribeau-Gesippe, Pour La Science, updated webpage, February 2017
• David Larousserie, Le Monde, February 22, 2017, Science & Médecine p.3, and webpage
• Edouard Thomas, Tangente, N°175, March-April 2017, p.8
• Olivier Lascar, Sciences et Avenir, updated webpage, March 10, 2017
• Philippe Fondanaiche, Diophante.fr, April 2017
• Jean-Paul Truc, Quadrature, N°106, Oct-Nov-Dec 2017, p.6