Highly multimagic squares: heptamagic, octamagic, and more...

In 2001, the most multimagic squares, known at that time, were the tetra and pentamagic squares (=4 and 5-multimagic) that I constructed with André Viricel. In 2003, Pan Fengchu constructed a hexamagic square (=6-multimagic) of order 4096 = 2¹². From 2004 to 2006, several new researches improved the multimagic records:

• in 2004 & 2005, by Pierre Tougne (PT), and by Pan Fengchu (PF)
• in 2005, by Harm Derksen, Christian Eggermont, Arno van den Essen (DEvdE), and by Jaroslaw Wroblewski (JW)
• in 2006, again Pan Fengchu (PF) from 9-multimagic to 14-multimagic

If I summarize all the new announced results, independently found:

Smallest known orders of highly multimagic squares

n-multimagic	square	PT	PF	DEvdE	JW
7	heptamagic	216	216	137	/
8	octamagic	221	221	178	/
9	nonamagic	223	223	179	/
10	decamagic	228	228	1910	229
11	hendecamagic	231	229	2311	no
12	dodecamagic	237	236	2312	no
13	tridecamagic	possible?	238	2913	no
14	tetradecamagic	possible?	244	2914	no
15 and +	pentadecamagic and +	possible?	possible?	yes	no

As I wrote in 2001 in my Pour La Science paper, page 101 (PDF file downloadable from the Bibliography page), the described "Viricel-Boyer" method used for the first tetra and pentamagic squares should be able to construct hexamagic, heptamagic, octamagic, and so on... up to any multimagic level, but with the need to increase sufficiently the size (= the order) of the square... and with the need to use a good computer program to check the squares... I am also sure, without having the proof, that the good old "Tarry-Cazalas" method is also able to create n-multimagic squares for any n.

Pierre Tougne used the two French methods. For example he constructed his heptamagic and octamagic squares with the Viricel-Boyer method, and the following keys:

• heptamagic square of order 2¹⁶
  row keys = (1, 367), column keys = (346, 17633)
• octamagic square of order 2²¹
  row keys = (1, 2539), column keys = (274, 1162381)

Pan Fengchu, China, also constructed numerous highly multimagic squares. I have not personally checked any of these huge squares announced by PF or PT.

In 2005, Harm Derksen (University of Michigan, USA), Christian Eggermont and Arno van den Essen (University of Nijmegen, Netherlands) wrote the first draft of a very interesting paper available here: http://arxiv.org/abs/math.CO/0504083. They proved mathematically that it is always possible to construct an n-multimagic square for any n. A very important theorem, supposed for a long time, announced by Gaston Tarry in 1906, but now proved!

However, the DEvdE method needs very big orders: for example, if we compare the heptamagic squares, their order 13^7 = 62,748,517 is approximatively one thousand times bigger than the order 2^16 = 65,536 used by Pierre Tougne and Pan Fengchu. I have not checked their heptamagic or higher squares: the squares should be correct because they are theoretically proved correct. But I asked Christian Eggermont to send me a trimagic square of the smallest possible order = 5^3 = 125 constructed with their method. I confirm that the received square has all the announced trimagic properties. An interesting fact, I noted that the Tarry-Cazalas method can be used to generate exactly the same square, with exactly the same cells.

An update: their "Multimagic squares" paper is now published in The American Mathematical Monthly, October 2007, pages 703-713.
I thank them for mentioning several times my works, but I regret that they don't mention that their multimagic square examples (the above trimagic square, but also their published bimagic squares) can be constructed by an old method first presented one century ago by Gaston Tarry in the Compte-Rendus de l’Académie des Sciences, and in more detail in Cazalas’s book. The reader may think that their construction method is fully new, and that's not true! Here are the Tarry-Cazalas keys of their published examples, pages 711-712:

• the example 1 of order 16 is a Tarry-Cazalas bimagic square using row keys (211, 97, 84, 168), column keys (21, 42, 199, 73), and additive key 40.
• the example 2 of order 9 is a Tarry-Cazalas bimagic square using row keys (34, 22), column keys (69, 37), no additive key.
• the example 3 of order 25 is a Tarry-Cazalas bimagic square using row keys (372, 318), column keys (194, 156), and additive key 102.
• and also the other example of order 25 given by Harm Derksen at http://www.math.lsa.umich.edu/~hderksen/magic.html with row keys (442, 282), column keys (156, 586), no additive key.

The authors were informed several months before the publication, but unfortunately they didn't want to add any remark on the Tarry-Cazalas method. Strange.

DEvdE proved that a n-multimagic square of order pⁿ exists for any prime p ≥ 2n − 1 with n ≥ 3. This result is improved with n=2 (bimagic squares) by Yong Zhang, Kejun Chen (Yancheng Teachers University, Jiangsu, China), and Jianguo Lei (Hebei Normal University, Hebei, China). In their paper "Large sets of orthogonal arrays and multimagic squares" published in the Journal of Combinatorial Designs, Vol.21, Issue 9, September 2013, pages 390-403 (but first published online in December 2012), they proved with their theorem 1.2:

a n-multimagic square of order pn exists for any prime p ≥ 2n − 1 with n ≥ 2

Jaroslaw Wroblewski (Wroclaw University, Poland) also worked on the subject of highly multimagic squares. Using Mathematica, he constructed a 10-multimagic square (=decamagic ≥square) of order 2^29, bigger than the order 2^28 used by Pierre Tougne or Pan Fengchu, but with a supplemental property: its 2^29 columns are 11-multimagic.

• Download "Magic squares J(p)", by Jaroslaw Wroblewski, Version 1 (Oct 16, 2005), PDF file of 77Kb

Return to the home page http://www.multimagie.com