A107739 Number of (completed) sudokus (or Sudokus) of size n^2 X n^2.
1, 1, 288, 6670903752021072936960
Offset: 0
Examples
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: 1 2 4 | 5 6 7 | 8 9 3 3 7 8 | 2 9 4 | 5 1 6 6 5 9 | 8 3 1 | 7 4 2 ------+-------+------ 9 8 7 | 1 2 3 | 4 6 5 2 3 1 | 4 5 6 | 9 7 8 5 4 6 | 7 8 9 | 3 2 1 ------+-------+------ 8 6 3 | 9 7 2 | 1 5 4 4 9 5 | 6 1 8 | 2 3 7 7 1 2 | 3 4 5 | 6 8 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
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
- D. Berend, On the number of Sudoku squares, Discrete Mathematics 341.11 (2018): 3241-3248.
- Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, The Stable Matching Problem and Sudoku, arXiv:2108.02654 [math.HO], 2021.
- Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, The Stable Marriage Problem and Sudoku, College Math. J. (2023).
- 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.
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
Comments