combinatorial question on 9x9

Showing 1-2 of 2 messages
combinatorial question on 9x9 QSCGZ 9/21/03 2:57 AM
Mark Brader wrote some weeks ago:

 >Some puzzles in the latest Games Magazine are based on a 9x9 array of
 >digits chosen from 1 to 9, with the following properties:
 >
 >   * each row contains each digit exactly once
 >   * each column contains each digit exactly once
 >   * if the 9x9 array is divided in thirds, forming a 3x3 array of
 >     3x3 subarrays, then each subarray contains each digit exactly once
 >
 >For example:
 >
 >      3 8 4   1 2 6   5 7 9
 >      9 2 6   5 7 3   4 1 8
 >      1 5 7   8 9 4   6 3 2
 >
 >      2 6 3   9 8 7   1 4 5
 >      5 7 1   4 6 2   8 9 3
 >      8 4 9   3 5 1   7 2 6
 >
 >      4 9 8   7 3 5   2 6 1
 >      7 3 2   6 1 8   9 5 4
 >      6 1 5   2 4 9   3 8 7

these are called "number place puzzles" .
They are popular in Japan where they are called "Sudoku" .

 >I have three questions that people might be interested in answering.
 >I don't know the answers myself.
 >
 >[1] How many distinct arrays are there that meet the conditions?

6670903752021072936960 = 9!*2^13*3^4*27704267971 = 6.67e21

 >[2] How many *essentially* distinct arrays -- as defined below --
 >    are there that meet the conditions?

about 2.8e9 , I don't know exactly.

 >[3] Does each of the essentially distinct arrays in [2] contribute the
 >    same number of distinct arrays to the total in [1], or not?

not. Most of them contribute 46656*2*2*6*6*9! , but not all.

 >Arrays are essentially distinct if it is NOT possible to generate one from
 >the other by a reasonably simple transformation -- either geometrical
 >(such as rotating the whole array) or numerical (such as complementing
 >the values in the whole array) -- or a combination of these.
 >--
 >Mark Brader   |  "...given time, a generally accepted solution to

I define two solved number place puzzles as equivalent , iff one can be
transformed into the other by a finite sequence of transformations ,
which include :
permuting the three rows 1,2,3
permuting the three rows 4,5,6
permuting the three rows 7,8,9
permuting the three 3*9-blocks consisting of rows  1+2+3,4+5+6,7+8+9
mirroring along the main diagonal
permuting the 9 symbols


--qscgz

combinatorial question on 9x9 Rammy 9/21/03 10:05 PM
*** post for FREE via your newsreader at post.newsfeed.com ***

>Mark Brader wrote some weeks ago:
>>puzzles ... based on a 9x9 array of digits

mark, may I ask why you are asking this?
is it pure mathematical curiousity?
or is there more?

R


 -----= Posted via Newsfeed.Com, Uncensored Usenet News =-----
http://www.newsfeed.com - The #1 Newsgroup Service in the World!
-----== 100,000 Groups! - 19 Servers! - Unlimited Download! =-----