Sudokus and bimagic squares

The Sudoku game has become a great success everywhere on the planet, mainly since 2005. The link between Latin squares and Sudokus has often been remarked, but it seems that the link between bimagic squares and Sudokus has never been remarked.

Gaston Tarry (Villefranche de Rouergue 1843 - Le Havre 1913)

In 1900, Gaston Tarry was the first to prove the famous problem of the 36 officers asked by Euler in 1782: it is impossible to arrange a delegation of six regiments (each of which sending a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant) in a 6x6 array such that no row or column duplicates a rank or a regiment. Tarry has also invented a marvelous method for the construction of multimagic squares : the first paper written by Tarry on his method was presented by Henri Poincaré in the Comptes-Rendus de l'Académie des Sciences, 1906. We can find details of his method, with numerous constructions of 8x8 and 9x9 bimagic squares using it, in General Cazalas's book published in 1934, the method slightly enhanced by Cazalas. And the Viricel-Boyer method used for our tetramagic and pentamagic squares, fully described in my article published in Pour La Science in 2001, is greatly inspired by the Tarry-Cazalas method.

Now the main information: ALL the 9x9 bimagic squares constructed with the Tarry-Cazalas method are a combination of 2 Sudokus! (remark made here in 2006, proved in 2011, see below)

Here are two Sudokus (each 3x3 subsquare contains the nine integers from 1 to 9, and each row and each column contains the nine integers from 1 to 9):

 Sudoku A Sudoku B 2 5 8 1 4 7 3 6 9 2 9 4 6 1 8 7 5 3 1 4 7 3 6 9 2 5 8 7 5 3 2 9 4 6 1 8 3 6 9 2 5 8 1 4 7 6 1 8 7 5 3 2 9 4 8 2 5 7 1 4 9 3 6 9 4 2 1 8 6 5 3 7 7 1 4 9 3 6 8 2 5 5 3 7 9 4 2 1 8 6 9 3 6 8 2 5 7 1 4 1 8 6 5 3 7 9 4 2 5 8 2 4 7 1 6 9 3 4 2 9 8 6 1 3 7 5 4 7 1 6 9 3 5 8 2 3 7 5 4 2 9 8 6 1 6 9 3 5 8 2 4 7 1 8 6 1 3 7 5 4 2 9

These Sudokus have nice supplemental properties, i.e. if we move one or more columns from one side to the opposite side, they remain Sudokus: the new 3x3 sub-squares contains again all the integers from 1 to 9. It is the same if we move rows from one side to the opposite side.

Now, construct a 9x9 square in which each cell uses the two cells of Sudokus A and B with the formula:

 9(A - 1) + B

Then you get a bimagic square constructed with the Tarry-Cazalas method!

 11 45 67 6 28 62 25 50 75 7 32 57 20 54 76 15 37 71 24 46 80 16 41 66 2 36 58 72 13 38 55 8 33 77 21 52 59 3 34 81 22 47 64 17 42 73 26 51 68 12 43 63 4 29 40 65 18 35 60 1 48 79 23 30 61 5 49 74 27 44 69 10 53 78 19 39 70 14 31 56 9

This square is bimagic:

• consecutive integers from 1 to 81.
• same sum S1 = 369 for each of the 9 rows, 9 columns and 2 diagonals
• after you have squared each number, the square remains magic, same sum S2 = 20049 for the 9 rows, 9 columns and 2 diagonals

And it has supplemental bimagic properties:

• again the same sum S1 = 369 for each of the nine 3x3 sub-squares (sum of the nine numbers in each sub-square)
• after you have squared each number, again the same sum S2 = 20049 for each of the nine 3x3 sub-squares (sum of the nine squared numbers in each sub-square)

You can get another bimagic square using the other formula:

 9(B - 1) + A

Of course, all pairs of Sudokus do not give a bimagic square, and all bimagic squares (those not constructed by the Tarry-Cazalas method) are not made from a couple of Sudokus. For example the first 9x9 bimagic square published by G. Pfeffermann cannot be constructed by the Tarry-Cazalas method, meaning that it is not a combination of 2 Sudokus.

Papers and books by Donald Keedwell

 In 2011, A. Donald Keedwell, Department of Mathematics, University of Surrey, England, (see his photo and other works here) published two interesting papers: "Gaston Tarry and multimagic squares", The Mathematical Gazette, Vol. 95, Number 534, November 2011, pp. 545-468 "Confirmation of a conjecture concerning orthogonal Sudoku and bimagic squares", Bulletin of the ICA (Institute of Combinatorics and its Applications), Volume 63, 2011, pp. 39-47 He explains Tarry-Cazalas's construction method of bimagic squares, mathematically proves my above remark of 2006 on the combination of 2 Sudokus in any Tarry's 9x9 bimagic square, and extends his remarks to p²xp² bimagic squares for every odd prime except five. < The cover of this issue of The Mathematical Gazette, with Gaston Tarry
 In 2015, Donald Keedwell published the second edition of his famous book "Latin Squares and their Applications" written with Jószef Dénes, foreword by Paul Erdös, first published in 1974. Pages 221 and 222, multimagic squares are defined, and Tarry's work is mentioned.Our work and website are referenced in several other pages: 205, 345, 346 (Dénes-Keedwell- Erdös problem 6.3 on add-mult squares), and 369.Thanks Donald! The cover of this second edition >"The three squares on the front cover of my book are mutually orthogonal latin squares and so form (are equivalent to) a projective plane of order 4. Thus, the cover of the 2nd edition is (almost) the same as the dust cover of the 1st edition. Two of the squares are (double) diagonal (maximum possible) and the third is unipotent and symmetric." Donald Keedwell, personal communication, Feb. 20, 2017

As we have seen above, Gaston Tarry is well known for his proof of the 36 officer problem of Euler, and for his construction method of multimagic squares. But also:

• in geometry, the Tarry point http://mathworld.wolfram.com/TarryPoint.html
• the Prouhet-Tarry-Escott problem http://mathworld.wolfram.com/Prouhet-Tarry-EscottProblem.html and numerous multigrades equations, for example
• the parametric equation, true from k = 0 to 5:
(6a - 3b - 8c)k + (5a - 9c) k + (4a - 4b -3c)k + (2a + 2b - 5c)k + (a - 2b + c)k + bk =
(6a - 2b - 9c)
k + (5a - 4b - 5c)k + (4a + b - 8c)k + (2a - 3b)k + (a + 2b - 3c)k + ck
• the equation, true from k = 0 to 10:
1k + 5k + 11k + 21k + 36k + 42k + 48k + 52k + 54k + 58k + 79k + 83k + 94k + 95k =
2
k + 3k + 14k + 18k + 39k + 43k + 45k + 49k + 55k + 61k + 76k + 86k + 92k + 96k
• the first known trimagic square (see the multimagic records)
• the inventor of the tetramagic term, using the greek root "tetra" instead of the latin root "quadra" or "quadri". It's in his memory that I used this term for the first known tetramagic square, built in 2001.
• collaboration on the two latest books written by Gabriel Arnoux, previously authored -without Tarry- of the first known pandiagonal perfect magic cube
• his works described in the Edouard Lucas's books:
• "Théorème des Carrefours" of Gaston Tarry, explained by Lucas in Théorie des Nombres (pages 107-109)
• "La Géométrie des Réseaux et le Problème des Dominos", a chapter fully dedicated to Gaston Tarry by Lucas in Récréations Arithmétiques (volume IV, 6th recreation, pages 123-151)
• "La Traversée des Ménages Modèles" and "La Traversée du Polygame", problems invented and solved by Gaston Tarry, presented by Lucas in L'Arithmétique Amusante (note II, pages 198-202)
• some of these works by Tarry are more briefly presented in Mathematical Recreations and Essays by W.W. Rouse Ball and H.S.M. Coxeter, and in Mathematical Recreations by Maurice Kraïtchik
• and numerous other things...

Nakamura-Taneja's bimagic square

Reorganizing cells of a magic (but not bimagic) square previously created by Mitsutoshi Nakamura piling up nine Sudoku Latin squares, Inder J. Taneja constructed this very nice 9x9 bimagic square: each number has 9 digits and contains all digits from 1 to 9, fun and astonishing! More details and squares in his paper http://rgmia.org/papers/v18/v18a159.pdf (square below page 11, result 22).

 123456789 297531864 345678912 486729153 561894237 618942375 759183426 834267591 972315648 459783126 534867291 672915348 723156489 897231564 945378612 186429753 261594837 318642975 786129453 861294537 918342675 159483726 234567891 372615948 423756189 597831264 645978312 378612945 156489723 231564897 642975318 429753186 594837261 915348672 783126459 867291534 615948372 483726159 567891234 978312645 756189423 831264597 342675918 129453786 294537861 942375618 729153486 894237561 315648972 183426759 267591834 678912345 456789123 531864297 264597831 312645978 189423756 537861294 675918342 453786129 891234567 948372615 726159483 591834267 648972315 426759183 864297531 912345678 789123456 237561894 375618942 153486729 837261594 975318642 753186429 291534867 348672915 126459783 564897231 612945378 489723156