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