cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

1, 1, 288, 6670903752021072936960
Offset: 0

Views

Author

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

Keywords

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.

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.

Crossrefs

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

A114288 Lexicographically earliest solution of any 9 X 9 sudoku, read by rows.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 5, 6, 7, 8, 9, 1, 2, 3, 7, 8, 9, 1, 2, 3, 4, 5, 6, 2, 1, 4, 3, 6, 5, 8, 9, 7, 3, 6, 5, 8, 9, 7, 2, 1, 4, 8, 9, 7, 2, 1, 4, 3, 6, 5, 5, 3, 1, 6, 4, 2, 9, 7, 8, 6, 4, 2, 9, 7, 8, 5, 3, 1, 9, 7, 8, 5, 3, 1, 6, 4, 2
Offset: 1

Views

Author

Zak Seidov, Nov 21 2005; corrected Nov 23 2005

Keywords

Comments

See A107739 for definition of a sudoku.

Examples

			1,2,3|4,5,6|7,8,9,
4,5,6|7,8,9|1,2,3,
7,8,9|1,2,3|4,5,6,
-----------------
2,1,4|3,6,5|8,9,7,
3,6,5|8,9,7|2,1,4,
8,9,7|2,1,4|3,6,5,
-----------------
5,3,1|6,4,2|9,7,8,
6,4,2|9,7,8|5,3,1,
9,7,8|5,3,1|6,4,2
		

Crossrefs

Programs

  • PARI
    A114288(m=3,n=3)={my(A=Mat([1..m*n]),L); for(i=2,m*n, A=matconcat([A; vector(#A, j, if(i%m!=1, A[i-1,(j-1+n)%#A+1], j>1||L=A[1,]; L=setminus(L, [j=setminus(L,Set(A[,j]))[1+(i==n+1&&j==m*n-1)]]); j))])); A} \\ Also allows to produce minimal sudokus of other sizes, e.g., 2x3 or 3x2. - M. F. Hasler, Nov 09 2021

A112454 Lexicographically maximal solution of any 9 X 9 sudoku, read by rows.

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 9, 8, 7, 3, 2, 1, 9, 8, 7, 6, 5, 4, 8, 9, 6, 7, 4, 5, 2, 1, 3, 7, 4, 5, 2, 1, 3, 8, 9, 6, 2, 1, 3, 8, 9, 6, 7, 4, 5, 5, 7, 9, 4, 6, 8, 1, 3, 2, 4, 6, 8, 1, 3, 2, 5, 7, 9, 1, 3, 2, 5, 7, 9, 4, 6, 8
Offset: 1

Views

Author

Chris Deugau (deugaucj(AT)uvic.ca), Dec 13 2005

Keywords

Comments

See A107739 for a definition of a sudoku.

Examples

			9 8 7|6 5 4|3 2 1
6 5 4|3 2 1|9 8 7
3 2 1|9 8 7|6 5 4
-----------------
8 9 6|7 4 5|2 1 3
7 4 5|2 1 3|8 9 6
2 1 3|8 9 6|7 4 5
-----------------
5 7 9|4 6 8|1 3 2
4 6 8|1 3 2|5 7 9
1 3 2|5 7 9|4 6 8
		

Crossrefs

A291188 Number of inequivalent 2 X n Sudoku grids ((2*n) X (2*n) grids divided into 2 X n boxes).

Original entry on oeis.org

2, 49, 1673187, 4743933602050718
Offset: 2

Views

Author

Seiichi Manyama, Aug 19 2017

Keywords

Comments

We regard two grids as inequivalent if one cannot be transformed into another by some sequence of symmetries, or by relabeling. For the 2 X 3 Sudoku grids case the allowed equivalences are (see link to Jarvis):
- relabeling entries; reflection; rotation;
- permutation of blocks of columns 1-2, 3-4 and 5-6;
- permutation of blocks of rows 1-3 and 4-6;
- permutation of columns 1-2;
- permutation of columns 3-4;
- permutation of columns 5-6;
- permutation of rows 1-3;
- permutation of rows 4-6.

Examples

			a(2) = 2:
.
  12|34  12|34
  34|12  34|12
  --+--  --+--
  21|43  23|41
  43|21  41|23
.
An example of 2 X 3 Sudoku grids:
.
  14|25|36
  35|16|24
  26|34|15
  --+--+--
  41|52|63
  52|63|41
  63|41|52
		

Crossrefs

A319900 a(n) is the number of distinct ways to arrange n copies of each of the numbers 1 through n^2 inside a fixed n X n X n cube, provided that no number appears twice in the same left-right plane, front-back plane, or top-bottom plane.

Original entry on oeis.org

1, 24, 14515200, 7708721243457872461824000
Offset: 1

Views

Author

Tanya Khovanova and Wayne Zhao, Sep 30 2018

Keywords

Comments

When n = 3, this is equivalent to enumerating the different fill-ins of a Sudo-Kurve puzzle of the shape given in the link 'Sudo-Kurve 38'.

Examples

			For n = 2, the top layer of the 2 X 2 X 2 cube must contain each of the numbers 1, 2, 3, 4. This can be arranged in 24 ways. Each way uniquely determines the rest of the cube, so there are 24 possible cubes.
		

Crossrefs

Formula

Observation: a(n) = A010791(n*(n-1)) for 1 <= n <= 3. - Omar E. Pol, Oct 02 2018

Extensions

a(4) from Bert Dobbelaere, Sep 20 2019
Showing 1-5 of 5 results.