I have seen the issue of the number of possible sudokus arise here and there, and so what I have to say has probably been said. I am fairly new to sudoku and have been thinking about this issue recently.

Apparently, the number of sudokus has been calculated, and it is a very large number. It seems to me, however, that if this very large number was determined merely through calculating the numer of possible arrangements of the numbers 1 through 9 that violate none of the rules of a properly constructed sudoku (each number appearing exactly once in each row, column and 3 x 3 box) then this very large number is not equivalent to the number of possible sudokus. I would submit that the actual number might be lower by many orders of magnitude. Consider:

As we know, a sudoku need not use numbers to fill the cells. One could use a group of 9 of anything--letters, shapes, colors, faces, words, objects, or technically even sounds, textures, smells, etc. Now if we were to take a completed sudoku and replace all the numbers with, say, colors, we would not say that we have created "another" sudoku--what we have done is merely to present the same sudoku in a different way. What makes a sudoku, then, is not the particular symbols or items we employ to express the sudoku but rather the set of logical relationships expressed by the chosen symbols or items.

The same reasoning holds, it seems, even when we don't switch the numbers of a given sudoku with non-numerical items but merely switch the numbers for each other--for instance, switching all the 7's for all the 3's in a sudoku. In such a case, the original sudoku and the modified sudoku exhibit the same exact set of logical relationships between the placeholders, it's just that the particular numbers that we use have been altered. In the accepted grand total of the number of possible sudokus, however, these "two" sudokus would be considered different sudokus. Thus the grand total involves some double-counting. The grand total counts every sudoku that looks different as different, but if I am right then sudokus can look different without being so. The grand total is in a sense an aesthetic total, not a logical total.

I have not sat down to figure out the number of possible ways in which the numbers in a sudoku could be switched about without altering the basic logical structure of the sudoku. My slapdash look hinted that there are 36 possible ways of swapping out one number for another, but of course one need not stop there--you could swap the 3's for the 8's, the 2's for the 4's, and the 5's for the 9's. You could do all sorts of swapping--36 is probably just the molecule on the tip of the iceberg.

Also, we should consider more than number-swapping. Take any given completed sudoku and place it on the table and give it a quarter-turn. Have we created a "new" sudoku? No, it seems not. We are merely looking at the same old sudoku from a slightly different angle. It is not the sudoku's fault that we changed the position of our chair relative to the sudoku. The grand total of sudokus, however, does count these as different sudokus. The same would apply when giving the sudoku a half-turn or a three-quarter-turn. Four persons sitting north, south, east, and west of a sudoku could all work on it simultaneously--but they are not working on 4 sudokus. Imagine that the table is made of glass. You could have 4 additional puzzledoers situated beneath the table--but this does not create 4 additional sudokus for a total of eight, but the grand total of possible sudokus does consider these to be 8 sudokus.

So for any sudoku you come across, there exists a large set of logically equivalent ways (using numbers--there exists an infinite set when we open the door to things other than numbers) of expressing that sudoku that are counted as different sudokus in the generally accepted grand total of possible sudokus. The question "How many sudokus are there?" is not merely a mathematical question but a logical one, and one best rephrased, "How many such sets of sudokus are there?" This number will itself be very large, but it will be smaller than the total arrived at mathematically.

One last thing: it would seem that for a casual puzzledoer the fact that one presentation of a sudoku differs from another would be sufficient for saying that they are "two different sudokus." Certainly they could be done in different ways (depending on how altered they are) without the puzzledoer noticing that they are equivalent. In fact, one could probably create an entire book of sudokus simply by turning, flipping and swapping out the numbers of one given sudoku, and most puzzledoers would not feel cheated by the book (although there might be a general sense of "this book is not as interesting as others I have done" or something like that) or notice its "laziness." For someone who is interested in the fundamental structural properties of these strange little logical beasties we call sudokus, however, any consideration that helps sweep away the superficial elements of sudokus is important.

I will end with a superficial element of sudokus: has anyone out there noticed that if you assign the position values 1 to 81 to the cells of a sudoku, the values of any given number in a completed sudoku add up to 369? Does this mean anything?