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A107739 Number of (completed) sudokus (or Sudokus) of size n^2 X n^2. 8
1, 1, 288, 6670903752021072936960 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

REFERENCES

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.

LINKS

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

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

Wikipedia, Sudoku

Eric Weisstein's World of Mathematics, Sudoku

EXAMPLE

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:

124|567|893

378|294|516

659|831|742

---+---+---

987|123|465

231|456|978

546|789|321

---+---+---

863|972|154

495|618|237

712|345|689

See A114288 for the lexicographically smallest 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

CROSSREFS

Cf. A108395, A109741, A114288, A198297.

Sequence in context: A008695 A047805 A173150 * A221203 A188367 A184046

Adjacent sequences:  A107736 A107737 A107738 * A107740 A107741 A107742

KEYWORD

nonn,bref,hard

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified November 28 06:47 EST 2014. Contains 250285 sequences.