MULTIMAGIE.COM - July 2008 news

List of the site's news added July 4, 2008

Publication of my two papers, with the most intriguing unsolved problems. Who can solve one (or several...) of them?

"Enigmes sur les Carrés Magiques" = "Enigmas on Magic Squares" in Dossier Pour La Science, Jeux Math, N°59 of April-June 2008. Thanks Philippe Boulanger and Bénédicte Leclercq, France.
Five enigmas, each with a 100€ prize + bottle of champagne!
"New Upper Bounds for Taxicab and Cabtaxi Numbers" in Journal of Integer Sequences, 2008, Volume 11, Issue 1. Thanks Jeffrey Shallit, Canada.
In part 8.3 is an unsolved problem equivalent to the above enigma 4: "Who can construct a 4x4 magic square of cubes?". See also http://www.cs.uwaterloo.ca/journals/JIS/vol11.html.

Computation of new numbers of multimagic series for squares by Michael Quist, USA: tetramagic and pentamagic series for squares of order 16, trimagic series for squares of order 15. He also confirmed the number of trimagic series for cubes of order 9 previously computed by Gildas Guillemot, France.
New page on the smallest tetramagic square problem: it is now proved that its order is ≥ 20 (and ≤ 243).
Interesting problem by Michael Cohen, USA, to appear in the Journal of Recreational Mathematics: find a bimagic square of order 5 in which all 12 row-column-diagonal sets of numbers are distinct sets.
Bimagic squares are know for any order ≤ 64! Thanks to Li Wen, China, with the first known bimagic squares of orders 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62 filling the gaps of the last unknown orders.
First known trimagic squares of orders 24 and 40 and second known trimagic square of order 16 by Li Wen, China (the first known of order 16 was constructed in 2005 by Chen Mutian and Chen Qinwu)
Numerous new results from Lee Morgenstern, USA:

First known 4x4 and 5x5 magic squares of pentagonal numbers, here is one of his two amazing 4x4 examples having the smallest possible magic sum (yes, smallest possible!):

His latest research on the 3x3 magic square of squares problem: complete formula for all 3x3 semi-magic squares of squares, and list of 3x3 semi-magic squares with 7 correct sums and using odd entries
Nice method looking for a 3x3 semi-magic square of cubes. Unfortunately... Uwe Hollerbach, USA, did not find any solution using his big list of numbers computed during his search on Cabtaxi(10): 3x3 semi-magic square of cubes still unknown.
There is no 7x7 semi-magic square of cubes using cubes between 1³ and 55³: 7x7 semi-magic squares of cubes still unknown.
Formulations of 3x3, 4x4, and 5x5 multiplicative magic squares
The magic product of a 6x6 pandiagonal multiplicative magic square is a 6th power.
New best known 6x6 add-mult semi-magic squares, but 6x6 add-mult magic squares still unknown.

a tetramagic square of order 256 was constructed by Charles Devimeux, France, in 1983... far before my tetramagic squares constructed in 2001 (with André Viricel) and 2003. So these can no longer be considered as the first tetramagic records...
on the first known bimagic cube of order 25, big doubts on the paternity of John R. Hendricks, Canada... This cube was more probably constructed by David M. Collison, USA.

Website of Christopher Henrich, USA, added in the links
Thanks (Michel Criton) for presenting in the Nov.-Dec. 2007 issue, page 2, my problem of the best possible multiplicative magic cube: still unsolved! This is also one of the problems later published in my Pour La Science paper of April 2008, paper mentioned at the beginning of this page.
Thanks (Maurice Mashaal) and (David Larousserie) for publishing in their January 2008 issues my solution of the problem proposed 67 years ago in The American Mathematical Monthly.