MULTIMAGIE.COM - Enigmas on Magic Squares

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!

Who can construct, or prove the impossibility:

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):

373²	289²	565²
360721	425²	23²
205²	527²	222121

5x5 bimagic square using distinct positive integers
3x3 semi-magic square of cubes using distinct positive cubed integers (small enigma #3a: square 7x7)
4x4 magic square of cubes using distinct positive cubed integers (small enigmas #4a, #4b, #4c: squares 5x5, 6x6, 7x7)
multiplicative magic cube using distinct positive integers < 364
5x5 additive-multiplicative magic square using distinct positive integers (small enigmas #6a , #6b: squares 6x6, 7x7)

No, I myself do not have the solutions... Of course, only the first person who will solve an enigma will win the associated prize and will have his name added in this table:

Recapitulative table of enigmas and of first discoverers.
The main enigma #5 does not appear here: different, it is the only one concerning magic cubes.
	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)	Impossible. Proved by L. Morgenstern, 2007
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 #3a (€100)	Small enigma #4c (€200)	Small enigma #6b (€200)
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), Germany (Trump), Poland (Wroblewski), USA (Horner, Morgenstern)

Enigmas in Pour La Science

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 on each of the 5 first main enigmas.
One year later in the Pour La Science website, the 5 same 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 offered on each of these six main enigmas, and six small enigmas from €100 to €500 are added.

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