MULTIMAGIE.COM - February 2009 news

List of the site's news added February 6, 2009

Numerous new additive-multiplicative magic squares (or semi-magic) from 10x10 to 25x25, with results from Su Maoting, China, Mohamad Moosavi, Iran, and me.
Magic when you add the cells, and again magic when you multiply the cells! Thanks to Eric W. Weisstein, Joshua Zucker and W. Edwin Clark, USA, for their independent checking of my squares.
36x36 pentamagic square using distinct -but not consecutive- integers by Li Wen, China. Congratulations, this is now (1) the smallest known pentamagic square using distinct integers, (2) the smallest known magic square of 5th powers, (3) and also the smallest known magic square of 4th powers!
Two first known 8x8 magic squares of cubes by Walter Trump, Germany. Congratulations, his two squares are the new smallest known magic squares of cubes, smaller than my 9x9!
So, now: who will construct a 7x7 magic square of cubes? (a difficult problem, see line just below...)
There is no 7x7 semi-magic square of cubes using any combination of 49 distinct cubes between 1³ and 58³, computed by Lee Morgenstern, USA. This means... 7x7 semi-magic or magic squares of cubes still unknown...
With the above new magic squares of powers, updated summary and tables. But the smallest possible magic squares of any power > 1 are still unknown...

Magic square	Smallest possible	Smallest known before ...	>>>>	Smallest known ...NOW!
of squares	Unknown! Maybe 3x3?	4x4 (Leonhard Euler, 1770)		4x4 (Leonhard Euler, 1770)
of cubes	Unknown! Maybe 4x4?	9x9 (Christian Boyer, 2006)		8x8 (Walter Trump, 2008)
of 4th powers	Unknown!	44x44 (J. Wrowlewski - H. Pfoertner, 2007)		36x36 (from the new pentamagic square of Li Wen, 2008)
of 5th powers	Unknown!	729x729 (from the pentamagic square of Li Wen, 2003)		36x36 (from the new pentamagic square of Li Wen, 2008)

There is no 6x6 bimagic square using 36 distinct integers < 72, computed by Lee Morgenstern, USA. This means that his 6x6 bimagic square with MaxNb = 72 can't be beaten!
Numbers of bimagic series for squares of orders 18, 19 and 20 computed by Michael Quist, USA. Very impressive computations!
Hexamagic series for squares of orders 17 and 19 are impossible, nicely proved by Jaroslaw Wroblewski, Poland.
If the work done by Michael Quist, USA, on multiplicative magic cubes is correct, the enigma #5 below is limited to the order 4, and to 221 ≤ Max nb < 364.

In spite of the above excellent recent results, the most intriguing problems are still unsolved! Who can solve one (or several...) of them, published in two journals in 2008:

In "Enigmes sur les carrés magiques", five enigmas, each with a 100€ prize + bottle of champagne!
In part 8.3 of "New Upper Bounds for Taxicab and Cabtaxi Numbers" , unsolved problem equivalent to the above enigma #4: "Who can construct a 4x4 magic square of cubes?".