Sudokus and bimagic squares
See also Sudoku's French ancestors

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):

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:

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

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:

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

About Gaston Tarry

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 + b^k =
  (6a - 2b - 9c)^k + (5a - 4b - 5c)^k + (4a + b - 8c)^k + (2a - 3b)^k + (a + 2b - 3c)^k + c^k
• the equation, true from k = 0 to 10:
  1^k + 5^k + 11^k + 21^k + 36^k + 42^k + 48^k + 52^k + 54^k + 58^k + 79^k + 83^k + 94^k + 95^k =
  2^k + 3^k + 14^k + 18^k + 39^k + 43^k + 45^k + 49^k + 55^k + 61^k + 76^k + 86^k + 92^k + 96^k
• 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).

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