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):

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!

Bimagic square constructed
from Sudoku A and Sudoku B

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:

And it has supplemental bimagic properties:

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 >

 


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:


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).

July 2015: Nakamura-Taneja's bimagic square
S1 = 4999999995 and S2 = 3376740371623259625
in 9 rows, 9 columns, 2 diagonals, and 9 sub-squares (3x3)

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


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