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):
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:
373² 
289² 
565² 
360721 
425² 
23² 
205² 
527² 
222121 
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:


2x2 
Impossible 

3x3 
Main enigma #1 
Impossible. Proved by E. Lucas, 1891 
Main enigma #3 
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 

5x5 
C. Boyer, 2004 
Main enigma #2 
C. Boyer, 2004 
Small enigma #4a 
Main enigma #6 
6x6 
C. Boyer, 2005 
J. Wroblewski, 2006 
L. Morgenstern, 2006 
Small enigma #4b 
Small enigma #6a 
7x7 
C. Boyer***, 2005 
L. Morgenstern, 2006 



8x8 
G. Pfeffermann***, 1890 
L. Morgenstern, 2006 
W. Trump, 2008 
W. Horner, 1955 

9x9 
G. Pfeffermann***, 1891 
L. Morgenstern  
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.
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. 









Who will be the next winner? With which enigma? 
Enigmas in Pour La Science... and elsewhere
in
Dossier Pour La Science (Jeux math')....... and
Pour La Science website
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:
With this Japanese paper in Sugaku Seminar, I know now that my name is written that way 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?
And thanks for their report of the solution of #4c found by Sébastien Miquel:
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