combinatorial question on 9x9

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QSCGZ

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Sep 21, 2003, 5:56:55 AM9/21/03
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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

Rammy

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Sep 22, 2003, 1:11:06 AM9/22/03
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>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


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evang...@gmail.com

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Apr 20, 2020, 4:59:05 AM4/20/20
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hey, almost 17 years...

CARLOCM

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Mar 17, 2021, 7:36:04 AM3/17/21
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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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