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%I #58 Oct 24 2023 12:39:56
%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 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 D. Berend, <a href="https://doi.org/10.1016/j.disc.2018.08.005">On the number of Sudoku squares</a>, Discrete Mathematics 341.11 (2018): 3241-3248.
%H Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, <a href="https://arxiv.org/abs/2108.02654">The Stable Matching Problem and Sudoku</a>, arXiv:2108.02654 [math.HO], 2021.
%H Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, <a href="https://doi.org/10.1080/07468342.2023.2261183">The Stable Marriage Problem and Sudoku</a>, College Math. J. (2023).
%H Bertram Felgenhauer and Frazer Jarvis, <a href="http://www.afjarvis.staff.shef.ac.uk/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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sudoku.html">Sudoku</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sudoku">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 1 2 4 | 5 6 7 | 8 9 3
%e 3 7 8 | 2 9 4 | 5 1 6
%e 6 5 9 | 8 3 1 | 7 4 2
%e ------+-------+------
%e 9 8 7 | 1 2 3 | 4 6 5
%e 2 3 1 | 4 5 6 | 9 7 8
%e 5 4 6 | 7 8 9 | 3 2 1
%e ------+-------+------
%e 8 6 3 | 9 7 2 | 1 5 4
%e 4 9 5 | 6 1 8 | 2 3 7
%e 7 1 2 | 3 4 5 | 6 8 9
%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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