%I A107739
%S A107739 1,1,288,6670903752021072936960
%N A107739 Number of (completed) sudokus (or Sudokus) of size n^2 X n^2.
%C A107739 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 A107739 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 A107739 Surendra Verma, The Little Book of Maths Theorems, Theories & Things, 
               New Holland Publishers (Australia) Pty Ltd., Sydney, page 135, 2008. 
               [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 30 2009]
%H A107739 Bertram Felgenhauer and Frazer Jarvis, <a href="http://www.shef.ac.uk/
               ~pm1afj/sudoku/">There are 6670903752021072936960 Sudoku grids</a>
%H A107739 J. P. Grossman, <a href="http://www.mathstat.dal.ca/~jpg/sudoku/">Javascript 
               Sudoku solver</a>
%H A107739 Ed Pegg Jr, <a href="http://www.maa.org/editorial/mathgames/mathgames_09_05_05.html">
               Sudoku variations</a>
%H A107739 Ed Russell and Frazer Jarvis, <a href="http://www.shef.ac.uk/~pm1afj/
               sudoku/sudgroup.html">There are 5472730538 essentially different 
               Sudoku grids</a>
%H A107739 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sudoku">Sudoku</a>
%H A107739 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Sudoku.html">Sudoku</a>
%e A107739 Comment from Hugo van der Sanden (hv(AT)crypt.org), 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 A107739 "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 A107739 "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 A107739 "Checking, we find just 6 of the 16 grids are valid:
%e A107739 1234/3412/2143/4321 1234/3412/2341/4123 1234/3421/2143/4312
%e A107739 1234/4312/2143/3421 1234/4321/2143/3412 1234/4321/2413/3142
%e A107739 so a(2) = 6 * 48 = 288."
%e A107739 An example of a sudoku of size 9 X 9:
%e A107739 124|567|893
%e A107739 378|294|516
%e A107739 659|831|742
%e A107739 ---+---+---
%e A107739 987|123|465
%e A107739 231|456|978
%e A107739 546|789|321
%e A107739 ---+---+---
%e A107739 863|972|154
%e A107739 495|618|237
%e A107739 712|345|689
%Y A107739 Cf. A109741.
%Y A107739 Cf. A108395. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 30 2009]
%Y A107739 Sequence in context: A159299 A047805 A008695 this_sequence A008367 A152852 
               A156572
%Y A107739 Adjacent sequences: A107736 A107737 A107738 this_sequence A107740 A107741 
               A107742
%K A107739 nonn,bref
%O A107739 0,3
%A A107739 Richard McNair (rmcnair(AT)ntlworld.com), Jun 11 2005
%E A107739 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Aug 12 2005
%E A107739 Thanks to Emiliano Venturini (il_wentu(AT)excite.com), for some corrections 
               to the comments, Apr 08 2006