MULTIMAGIE.COM - March 2012 news

List of the site's news added March 8th, 2012

NEW RECORDS! Four new smallest known semi-magic or magic squares of powers!

15x15 magic square of 4th powers by François Labelle and Christian Boyer, smaller than Labelle's 16x16 (the previous record)
16x16 semi-magic square of 5th powers, by Jaroslaw Wroblewski, smaller than Boyer's 25x25 (the previous record)
36x36 semi-magic square of 6th powers, by Jaroslaw Wroblewski
90x90 magic square of 6th powers, by François Labelle, much smaller than Pan Fengchu's 4096x4096 (the previous record)

With the new results above, here is the updated table of smallest known squares of powers:

Edouard Lucas	Smallest known squares	semi-magic	magic	Leonhard Euler
	of squares	3x3, Edouard Lucas (1876)	4x4, Leonhard Euler (1770)
	of cubes	4x4, Lee Morgenstern (2006)	8x8, Walter Trump (2008)
	of 4th powers	4x4, Morgenstern-Boyer (2006)	15x15, Labelle-Boyer (2011) NEW!
	of 5th powers	16x16, J. Wroblewski (2011) NEW!	36x36, Li Wen (2008)
	of 6th powers	36x36, J. Wroblewski (2011) NEW!	90x90, François Labelle (2011) NEW!

But smallest possible squares not yet known! Who will find smaller squares? Only 3x3 semi-magic squares of squares are proved impossible to improve (2x2 being of course impossible!).
The goal of several enigmas (#1, #3, #4, #4a, #4b, #4c) is to improve this table.

There still remain ten enigmas for winning €6,900 and ten bottles!
Here are the most interesting results received on the remaining enigmas, but still unsolved:

#1 (3x3 magic square using at least 7 distinct squared integers), interesting contributions from Ant King, Lee Morgenstern and Mike Winkler
#2 (5x5 bimagic square), no solution using integers < 300 with 23 correct sums found by Gildas Guillemot (a solution to this enigma needs 24 correct sums)

On bimagic squares:

Two papers published by Donald Keedwell, in The Mathematical Gazette and in The Bulletin of the ICA, proving my remark of 2006:

ALL the 9x9 bimagic squares constructed with the Tarry-Cazalas method are a combination of 2 Sudokus!

"Gaston Tarry and multimagic squares" on the cover of the Nov. 2011 issue of The Mathematical Gazette!
(title of the paper by Donald Keedwell)

Paper published by Chen Kenju, Li Wen, and Pan Fengchu in the Journal of Combinatorial Designs:

pandiagonal bimagic squares of order n⁴ exist for all integer n ≥ 7 such that gcd(n,30) = 1

Paper published by Li Wen, Wu Dianhua, and Pan Fengchu in Discrete Mathematics:

pandiagonal bimagic squares of order (pq)ⁿ exist for n ≥ 1 and (p, q) = (11, 7), (13, 7), (19, 7), (13, 11), or (17, 11)

Pandiagonal bimagic squares of orders 32, 40, 56, 64, 72, ... constructed by Pan Fengchu
Enumeration of Tarry's 8x8 bimagic squares (and pandiagonal) by Francis Gaspalou
There are no compact bimagic squares of order 8 proved by Francis Gaspalou, and of any order proved by Walter Trump
Complete parametric solutions of 4x4 semi-bimagic squares by Lee Morgenstern, but no solution found of 19 correct sums

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