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A107739 Number of (completed) sudokus (or Sudokus) of size n^2 X n^2. 13


%S 1,1,288,6670903752021072936960

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

%C 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.

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

%D K. Ying Lin, "Number Of Sudokus" in 'Journal of Recreational Mathematics' pp. 120-4 Vol.33 No. 2 2004-5 Baywood Pub. Amityville NY.

%D Surendra Verma, The Little Book of Maths Theorems, Theories & Things, New Holland Publishers (Australia) Pty Ltd., Sydney, page 135, 2008.

%H Bertram Felgenhauer and Frazer Jarvis, <a href="http://www.shef.ac.uk/~pm1afj/sudoku/">There are 6670903752021072936960 Sudoku grids</a>

%H J. P. Grossman, <a href="http://www.mathstat.dal.ca/~jpg/sudoku/">Javascript Sudoku solver</a>

%H Ed Pegg Jr, <a href="http://www.maa.org/editorial/mathgames/mathgames_09_05_05.html">Sudoku variations</a>

%H Ed Russell and Frazer Jarvis, <a href="http://www.afjarvis.staff.shef.ac.uk/sudoku/sudgroup.html">There are 5472730538 essentially different Sudoku grids</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sudoku">Sudoku</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sudoku.html">Sudoku</a>

%H Krasimir Yordzhev, <a href="http://doi.org/10.1016/j.disc.2016.09.011">On the number of mutually disjoint pairs of S-permutation matrices</a>, Discrete Mathematics 340 (2017) 1442-1448.

%e 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).

%e "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.

%e "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.

%e "Checking, we find just 6 of the 16 grids are valid:

%e 1234/3412/2143/4321 1234/3412/2341/4123 1234/3421/2143/4312

%e 1234/4312/2143/3421 1234/4321/2143/3412 1234/4321/2413/3142

%e so a(2) = 6 * 48 = 288."

%e An example of a sudoku of size 9 X 9:

%e 124|567|893

%e 378|294|516

%e 659|831|742

%e ---+---+---

%e 987|123|465

%e 231|456|978

%e 546|789|321

%e ---+---+---

%e 863|972|154

%e 495|618|237

%e 712|345|689

%e 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

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

%K nonn,bref,hard

%O 0,3

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

%E Entry revised by _N. J. A. Sloane_, Aug 12 2005

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

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)