? How many different Sudoku matrices can be formed? Would anyone check out my logic below?
I count for the first row. 362880 permutations
The next row has a smaller number due to the restriction of 1-9 in each box.
The second row has 3 of 6 number remaining *
3 of 6 numbers remaing * 3 of 3 remaing.
The 3rd row is further restricted..
to 6 x6 x 6.. the remaining permutations..

The next set of boxes column by column
120,*6*6..
Leaving lastly the final row of boxes.
6x6x6 ..
The remaining boxes are fully defined.
I get an estimate 6.3193E18 permutations
of completed Sudoku puzzles.

The starting puzzles with blank spots will of course be much much greater.

Googling around, I found two sources that give the same number:

(1) Enumerating Possible Sudoku Grids (.pdf) by Felgenhauer and Jarvis

(2) A message board called Wilmott Forums with a topic titled "How many Sudoku - 9x9 Square"

They give 6,670,903,752,021,072,936,960 (≈ 6.67×10²¹) = (9!)(72²)(2⁷)(27,704,267,971) unique grids (the last factor being prime).

However, ignoring reflections, rotations, and other "degeneracies" of symmetries in the grids eliminates (9!)(72²), leaving "only" 3,546,146,300,288 (≈ 3.5x10¹²) arrangements.

I confess I didn't follow the analysis and logic in detail, but it's all there in those links for anyone to study. It's a non-trivial calculation and apparently requires hours of computing time once it's properly set up.