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:

 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.