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 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


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:


Return to the home page http://www.multimagie.com