Tetramagic squares and pentamagic squares

 Warning! The tetramagic squares below can no longer be considered the first constructed! Look at the 256x256 tetramagic square constructed in 1983 by Charles Devimeux.But the pentamagic squares are (or still seem to be?) the first constructed.

André Viricel (Nancy 1913 - Villers lès Nancy 2003)

With my friend André Viricel, we have had the honour to construct in May 2001 the first known tetramagic square (4-multimagic). Its order is 512 (so is 512x512 in size). And one month later, in June 2001, we beat our own record by constructing the first know pentamagic square (5-multimagic). Its order is bigger: 1024 (so is 1024x1024 in size).

These 2 squares were the subject of a detailed article published in the August 2001 issue of Pour La Science, the French edition of Scientific American, and included an explanation of the constructing method used.

Before we announced these squares, we asked several scientists to verify them, using separate software programs, it is necessary to be prudent! Many thanks to Yves Gallot (Toulouse, C and Proth's Integer library), Renaud Lifchitz (Paris, Mathematica), and Olivier Ramare (Lille, MuPAD).

You can also verify by yourselves in donwloading the full record squares:

Know the features of your verification program. It has to know how to calculate in multiprecision, since the pentamagic sum of each line (row, column or diagonal) of the 1024th-order square is around 2x10^32.

 Why TETRA-magic, and not QUADRA or QUADRI-magic? Gaston Tarry has been the first to imaginate these 4-multimagic squares. He had then preferred the "tetra"magic term using the Greek root rather than for example "quadri"magic using the Latin root. He has used for the first time this term of "tetra"magic in 1905, at the time of the official presentation of his trimagic square of order 128 (see Association Française pour l'Avancement des Sciences, 1905, page 35). So we have kept this terminology in his memory. And going on to use the Greek, and not the Latin, for other squares: "penta", "hexa",...

Here are the four corners of our two squares:

 0 139938 18244 … 243899 122205 262143 140551 1957 156227 … 105916 260186 121592 18959 157869 3403 … 258740 104274 243184 … … … … … … … 242703 104109 258891 … 3252 158034 19440 121607 260517 105539 … 156604 1626 140536 261632 122018 244036 … 18107 140125 511

 A tetramagic square of order 256 has been constructed in January 2003, so smaller than the above tetramagic square. The number 2003 is the first number of the square, so located in the high left corner. Its multimagic sums are: S1  = 8388480 S2  = 366495487360 S3  = 18013848757862400 S4  = 944437268143413954688 Excel being able to work with sheets until 256 columns, this square can now be stored in an Excel file, contrary to the previous tetramagic square. It can be downloaded here:      tetramagic square of order 256 (zipped Excel file 206Kb) constructed by Christian Boyer.

 A tetramagic square of order 243 has been constructed in February 2004, so again smaller than the above tetramagic square. By Pan Fengchu, China. Its multimagic sums are: S1  = 7174332 S2  = 282422362068 S3  = 12507462634032432 S4  = 590837525820046776348 It can be downloaded here:      tetramagic square of order 243 (zipped Excel file 189Kb) constructed by Pan Fengchu.

 0 733632 419712 … 628863 314943 1048575 866545 395569 745329 … 303246 653006 182030 685538 82978 791138 … 257437 965597 363037 … … … … … … … 685597 83933 790941 … 257634 964642 362978 867086 395982 744590 … 303985 652593 181489 1023 733759 418943 … 629632 314816 1047552

And here is the integral list of the properties of this pentamagic square:

• each number from 0 to 1,048,575 is present once and only once... if not, it would be very easy to build magic squares ;-)
• the square is magic: the sums of the numbers of each of the 1024 rows, 1024 columns and 2 diagonals are all equal to S1
• the square is bimagic: the sums of the squared numbers of each of the 1024 rows, 1024 columns and 2 diagonals are all equal to S2
• the square is trimagic: the sums of the cubed numbers of each of the 1024 rows, 1024 columns and 2 diagonals are all equal to S3
• the square is tetramagic: the sums of the 4th-powered numbers of each of the 1024 rows, 1024 columns and 2 diagonals are all equal to S4
• the square is pentamagic: the sums of the 5th-powered numbers of each of the 1024 rows, 1024 columns and 2 diagonals are all equal to S5

The magic sums S1 to S5 are:

• S1 =    536870400
• S2 =    375299432076800
• S3 =    295147342229667840000
• S4 =    247587417561640996243120640
• S5 =    216345083469423421673932062720000

The presence of 0 bothers you? That is easy to solve: simply add 1 to all numbers, and the square retains the same multimagic properties (the S1 to S5 values are of course changed).

The scientific French press talked about our records: Pour La Science of August 2001, Sciences et Avenir of September 2001, Science et Vie of November 2001. But it is evident that our 2 records are pale compared to the first multimagic squares built by hand. Today, computing makes the calculations very much simpler, but makes for less merit for the record holder.

 A pentamagic square of order 729 has been constructed in June 2003, so smaller than the above pentamagic square. See the description of this square constructed by Li Wen, in China.

We have had the pleasure that our pentamagic square has been officially recognized in May 2002 by the Guinness World Records. Here is the nice diploma:

If you do not understand french, "Découverte du premier carré pentamagique connu" means "Discovery of the first known pentamagic square" and not the "Most multimagic square". So we are sure that, even if somebody find an hexamagic square, the diploma will remains valid ;-)

 During the night of August 14th-15th 2003, our friend André Viricel passed away in his home located in the suburb of Nancy (France). He will never read the dedication specially written for him in the article of Pour La Science published in September 2003, page 94. He had a passion for various mathematical subjects, including magic squares. He was a regular writer in Le Petit Archimède, an old and famous French magazine about recreational mathematics published by the ADCS association in Amiens. This mathematical teacher, graduated from the famous Ecole Normale Supérieure of Saint-Cloud, was a brilliant mind. And always so affable. We are sad.Read the eulogy to André Viricel, written (in French) by Yves Roussel, president of the ADCS.