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Number of "magic sudokus" (and random generation)

 
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DrSudoku

Joined: 30 Oct 2005
Posts: 1
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PostPosted: Sun Oct 30, 2005 12:17 pm    Post subject: Number of "magic sudokus" (and random generation) Reply with quote

Hi,

I am intereste in calculating how many "magic sudokus" it's possible to create. A "magic sudoku" is a normal sudoku with an extra constraint:
in each of the 9 3x3 squares the sum of each row and col must be the same (15).

I have read about the difficulty in calculating the number of regular sudokus but I reckon it's possible to calculate the numbers when it's a magic sudoku. The number of possible 3x3 squares where the sum of each row and col is 15 should be each to calculate.

I have written a simple program i c to generate random "magic sudokus" but I am interested in knowing how many it's possible to make. I have a hunch that it's not that many.

Here's some output examples from my program:

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


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


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

Thanks in advance.
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dukuso

Joined: 14 Jul 2005
Posts: 454
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Location: germany

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PostPosted: Sun Oct 30, 2005 2:14 pm    Post subject: Reply with quote

I think it's
Code:

5.7.9..


someone else please fill in the missing digits so to confirm it.
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mrcl

Joined: 01 Dec 2005
Posts: 3
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PostPosted: Thu Dec 01, 2005 12:54 am    Post subject: Reply with quote

dukuso wrote:
I think it's
Code:

5.7.9..


maybe 5971968

besides transposition and reordering column or rows the only possible box of such a sudoku is
Code:
159
672
834

transposition divides the boxes into two classes. boxes of different classes cannot be in the same sudoku. there are 2*3!*3! posibilities to select a box 1: transposition (2) ,permuting rows (3!) or columns (3!).
for box2 there are 2!*3! possibilites: 2! for permuting the rows and 3! for permuting the columns. all in all there are the following possibilities to seleect boxes:

Code:

  2*3!*3!  3!*2!  3!*1!
    2!*3!  2!*2!  2!*1!
    1!*3!  1!*2!  1!*1!

this gives 2*2!^6*3!^6 possiblitis, that gives 5971968.
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