A114288 -id:A114288 - cp's OEIS frontend

A107739 Number of (completed) sudokus (or Sudokus) of size n^2 X n^2.

1, 1, 288, 6670903752021072936960

Offset: 0

Richard McNair (rmcnair(AT)ntlworld.com), Jun 11 2005

An n^2 X n^2 sudoku is an n^2 X n^2 array which is subdivided into n^2 n X n subarrays. Each row and column of the full array must contain each of the numbers 1 ... n^2 exactly once (this makes it a Latin square of order n^2). In addition, each of the n^2 n X n subarrays must also contain each of the numbers 1 ... n^2 exactly once.

			Comment from _Hugo van der Sanden_, Jun 12 2005: "Consider n=2: renumbering doesn't affect the result, so we can fix row A at (1, 2, 3, 4) and multiply the result by 4!. Once rows B and C are chosen, there is only one option for row D. Row B must have (3, 4) or (4, 3) followed by (1, 2) or (2, 1).
"Rows C and D can be swapped without affecting validity, so we can fix column 1 of row C to be the lower of the two options and multiply the results by 2.
"That leaves at most 4 options for row C (2 choices in each of the remaining 3 positions, of which one must have our selected number as one of the choices); that leaves 16 options to check for rows B and C, the result to be multiplied by 48.
"Checking, we find just 6 of the 16 grids are valid:
1234/3412/2143/4321 1234/3412/2341/4123 1234/3421/2143/4312
1234/4312/2143/3421 1234/4321/2143/3412 1234/4321/2413/3142
so a(2) = 6 * 48 = 288."
An example of a sudoku of size 9 X 9:
  1 2 4 | 5 6 7 | 8 9 3
  3 7 8 | 2 9 4 | 5 1 6
  6 5 9 | 8 3 1 | 7 4 2
  ------+-------+------
  9 8 7 | 1 2 3 | 4 6 5
  2 3 1 | 4 5 6 | 9 7 8
  5 4 6 | 7 8 9 | 3 2 1
  ------+-------+------
  8 6 3 | 9 7 2 | 1 5 4
  4 9 5 | 6 1 8 | 2 3 7
  7 1 2 | 3 4 5 | 6 8 9
See A114288 for the lexicographically earliest 9 x 9 solution, which is the analog of the first of the 4 x 4 grids given at the end of van der Sanden's comment. - _M. F. Hasler_, Mar 29 2013

K. Ying Lin, "Number Of Sudokus" in 'Journal of Recreational Mathematics' pp. 120-4 Vol.33 No. 2 2004-5 Baywood Pub. Amityville NY.
Surendra Verma, The Little Book of Maths Theorems, Theories & Things, New Holland Publishers (Australia) Pty Ltd., Sydney, page 135, 2008.

D. Berend, On the number of Sudoku squares, Discrete Mathematics 341.11 (2018): 3241-3248.
Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, The Stable Matching Problem and Sudoku, arXiv:2108.02654 [math.HO], 2021.
Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, The Stable Marriage Problem and Sudoku, College Math. J. (2023).
Bertram Felgenhauer and Frazer Jarvis, There are 6670903752021072936960 Sudoku grids
J. P. Grossman, Javascript Sudoku solver
Ed Pegg Jr, Sudoku variations
Ed Russell and Frazer Jarvis, There are 5472730538 essentially different Sudoku grids
Eric Weisstein's World of Mathematics, Sudoku
Wikipedia, Sudoku
Krasimir Yordzhev, On the number of mutually disjoint pairs of S-permutation matrices, Discrete Mathematics 340 (2017) 1442-1448.

Cf. A108395, A109741, A114288, A198297, A285178-A285180.

Entry revised by N. J. A. Sloane, Aug 12 2005

Thanks to Emiliano Venturini (il_wentu(AT)excite.com), for some corrections to the comments, Apr 08 2006

A109741 Number of inequivalent (completed) n^2 X n^2 sudokus (or Sudokus).

Original entry on oeis.org

1, 1, 2, 5472730538

Offset: 0

N. J. A. Sloane, Aug 13 2005

See A107739 for definition of an n^2 X n^2 sudoku.
a(2) = 2 independently computed by Gary McGuire and Hugo van der Sanden.
For the 9 X 9 case the allowed equivalences are (see link to Jarvis et al.):
- relabeling entries; reflection; rotation;
- permutation of blocks of columns 1-3, 4-6 and 7-9;
- permutation of blocks of rows 1-3, 4-6 and 7-9;
- permutation of columns 1-3; permutation of rows 1-3;
- permutation of columns 4-6; permutation of rows 4-6;
- permutation of columns 7-9; permutation of rows 7-9.
See A107739 for the total number of ("square") sudoku grids, A114288 for the lexicographically earliest 9 X 9 solution. - M. F. Hasler, Mar 29 2013

			a(2) = 2:
12|34 12|34
34|12 34|12
--+-- --+--
21|43 23|41
43|21 41|23

J.-P. Delahaye, "Le tsunami du Sudoku" in 'Pour La Science' (French Edition of "Scientific American"), December 2005 pp. 144-9, Paris.

J. P. Grossman, Javascript Sudoku solver
S. Gupta, Exploring the Mathematics of Su Doku
Frazer Jarvis et al., Sudoku enumeration problems
Eric Weisstein's World of Mathematics, Sudoku
Wikipedia, Sudoku

Links to Jarvis et al. fixed by M. F. Hasler, Mar 29 2013

A112454 Lexicographically maximal solution of any 9 X 9 sudoku, read by rows.

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 9, 8, 7, 3, 2, 1, 9, 8, 7, 6, 5, 4, 8, 9, 6, 7, 4, 5, 2, 1, 3, 7, 4, 5, 2, 1, 3, 8, 9, 6, 2, 1, 3, 8, 9, 6, 7, 4, 5, 5, 7, 9, 4, 6, 8, 1, 3, 2, 4, 6, 8, 1, 3, 2, 5, 7, 9, 1, 3, 2, 5, 7, 9, 4, 6, 8

Offset: 1

Chris Deugau (deugaucj(AT)uvic.ca), Dec 13 2005

See A107739 for a definition of a sudoku.

			9 8 7|6 5 4|3 2 1
6 5 4|3 2 1|9 8 7
3 2 1|9 8 7|6 5 4
-----------------
8 9 6|7 4 5|2 1 3
7 4 5|2 1 3|8 9 6
2 1 3|8 9 6|7 4 5
-----------------
5 7 9|4 6 8|1 3 2
4 6 8|1 3 2|5 7 9
1 3 2|5 7 9|4 6 8

Cf. A107739, A109741, A114288.

A211172 A Sudoku torus.

Original entry on oeis.org

6, 5, 7, 3, 4, 2, 1, 9, 8, 9, 8, 1, 5, 6, 7, 4, 3, 2, 3, 2, 4, 8, 9, 1, 6, 5, 7, 5, 7, 6, 2, 3, 4, 9, 8, 1, 8, 1, 9, 7, 5, 6, 3, 2, 4, 2, 4, 3, 1, 8, 9, 5, 7, 6, 7, 6, 5, 4, 2, 3, 8, 1, 9, 1, 9, 8, 6, 7, 5, 2, 4, 3, 4, 3, 2, 9, 1, 8, 7, 6, 5

Offset: 1

Jon Perry, Jan 31 2013

The sequence is a listing for a Sudoku grid:
6 5 7 3 4 2 1 9 8
9 8 1 5 6 7 4 3 2
3 2 4 8 9 1 6 5 7
5 7 6 2 3 4 9 8 1
8 1 9 7 5 6 3 2 4
2 4 3 1 8 9 5 7 6
7 6 5 4 2 3 8 1 9
1 9 8 6 7 5 2 4 3
4 3 2 9 1 8 7 6 5
No two diagonally adjacent elements are the same. If the grid is rolled into a cylinder either way, this is still true making a 'Sudoku torus'.
This extra information can be used to construct puzzles that use 'no diagonal' logic.
The two lexicographical Sudoki in the cross-references have one internal diagonal each and several external diagonals.

			If a puzzle has say:
   |x
  5|
___|x
then x cannot be 5.

Mathworld, Sudoku
Wikipedia, Sudoku

Cf. A107739.

Cf. A114288, A112454 (lexicographic grids).

cp's OEIS Frontend

A107739 Number of (completed) sudokus (or Sudokus) of size n^2 X n^2.

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A109741 Number of inequivalent (completed) n^2 X n^2 sudokus (or Sudokus).

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A112454 Lexicographically maximal solution of any 9 X 9 sudoku, read by rows.

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A211172 A Sudoku torus.

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A175547 Lexicographically earliest sequence of increasing 9-digit zeroless anagrams that share no common digit place with previous terms.

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