What makes a unique sudoku?

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I've been trying to figure out how many different n x n "grids without subgrids" are possible, and I came across something that seemed quite counter-intuitive to me.

I'd assumed that to find, for instance, the number of possible 6x6 grids, you'd start with the number of possible permutations for the first row ( 6! = 720 ). Then you'd choose an arbitrary permutation for the first row, and work out how many possibilities that left for the second row (it seems to be 265). Then you'd arbitrarily choose on of those permutations for the second row and then move on to the third row, and so on. Finally, to find the number of possible 6x6 grids, you'd multiply together the numbers for each row ( 720 x 265 x ... etc).

But it seems that the results can be different depending on which choices you make. For example, if you have the following arrangement for the first four rows:

1 2 3 4 5 6
2 3 4 5 6 1
3 4 5 6 1 2
4 5 6 1 2 3

This leaves only two possibilities for the fifth row, namely:

5 6 1 2 3 4 ... or ...
6 1 2 3 4 5

But if you instead have the following arrangement for the first four rows:

1 2 3 4 5 6
2 1 4 3 6 5
3 4 5 6 1 2
4 3 6 5 2 1

You now have eight possibilities for the fifth row:

5 6 1 2 3 4 ... or ...
5 6 1 2 4 3 ... or ...
5 6 2 1 3 4 ... or ...
5 6 2 1 4 3 ... or ...
6 5 1 2 3 4 ... or ...
6 5 1 2 4 3 ... or ...
6 5 2 1 3 4 ... or ...
6 5 2 1 4 3

Maybe it's just me, but this was kind of unexpected, although it makes some sort of sense when I compare the two arrangements.

So does anyone know how many 6x6 grids of this type are actually possible?
(And I used a computer program to help me with this!)

It gets messy very quickly: http://en.wikipedia.org/wiki/Latin_square

I believe that the editors' personality subtly comes through in which puzzles they select for publication. Will Shortz' Sudokus , the diifficult ones at least, have a slightly different feel than those of other authors. The difficult ones in his second book felt more complex than ones you would see on websites and newspapers. He may have run out of good ones because some of the final dificult ones in his third book were easy and reminded me of others'.

Anxiled: The link you gave in another thread with the solving strategies answers my questions. From what the programmers are discussing in the forum on that site, it is clear that the programs to produce unique puzzles do in fact add numbers one by one and try to solve it after each one. I guess if your solving algorithm is fast at determining whether the solution is unique, this is not a problem.

Also, apparently there are algorithms that imitate a human trying to solve a puzzle. These are applied once a unique puzzle is already generated, and give a score to each move depending on how easy it is. This allows an approximate difficulty level to be assigned automatically, though the only true test is to have people actually try to solve it.