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 lieutenantcolonel, a major, a captain, a lieutenant, and a sublieutenant) 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 ComptesRendus 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 ViricelBoyer 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 TarryCazalas method.
Now the main information: ALL the 9x9 bimagic squares constructed with the TarryCazalas 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 subsquares 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 TarryCazalas 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:
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 TarryCazalas 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 TarryCazalas 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:
He explains TarryCazalas'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. The cover of this second edition > 
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:
NakamuraTaneja'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 
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