How many distinct Sudoku puzzles are there?

Question

Can the total number of distinct Sudoku puzzles be calculated? Any experts in math stat here? How in the world would you calculate this? I was a math major many years ago, but I can't quite figure it out. Please post the solution if you can.

Answer 1

That's a tough one, and I don't have an answer.

Suppose you started filling in the grid by starting in the upper left hand corner, proceeding across the first row, then going left to right for the second row, then the third row, and so on.

There would be 9! possibilities for the first row.
There would be only one possibility for the whole last row.
There would be at most 2 possibilities each for each entry in the second to last row.
There would be at most 6 possibilities each for ANY entry after the first row (think about it).
There would be at most 3 possibilities each for any entry in the 3rd or 6th rows.
There would be only one possibility each for the entries in the rightmost column, at most 2 possibilities each for the entries in the second rightmost column, and so on.

But all that is a whole lot of "at mosts". I don't immediately see how to narrow it down more precisely.