• #### combinatorial question on 9x9

From evangallade@gmail.com@21:1/5 to All on Mon Apr 20 01:59:04 2020
hey, almost 17 years...

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• From CARLOCM@21:1/5 to All on Wed Mar 17 04:36:03 2021
El domingo, 21 de septiembre de 2003 a las 11:56:55 UTC+2, QSCGZ escribiÃ³:
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

Hey!
I was just wondering why it's that number. I tried with the 3 and 4 sudokus and this formula gave it the result perfectly: 3! * 2! *1! (=12) and 4! * 3! * 2! * 1! = 288

If I just do that until 9 I end up with this: 9! * 8! * 7! * 6! * 5! * 4! * 3! * 2! = 7,34*10^21 which is suprisingly close to your number. I don't know where your numbers come from...

Thank you!

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