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A107739 Number of (completed) sudokus (or Sudokus) of size n^2 X n^2. 13
1, 1, 288, 6670903752021072936960 (list; graph; refs; listen; history; text; internal format)



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.


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.


Table of n, a(n) for n=0..3.

D. Berend, On the number of Sudoku squares, Discrete Mathematics 341.11 (2018): 3241-3248.

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.


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:












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


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

Sequence in context: A296608 A306651 A272164 * A221203 A188367 A184046

Adjacent sequences:  A107736 A107737 A107738 * A107740 A107741 A107742




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


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



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Last modified October 20 13:54 EDT 2020. Contains 337904 sequences. (Running on oeis4.)