Sudoku: Ah the Possibilities

Recently my roommate and I were wondering how many different Sudoku puzzles it is possible to make. We first figured out one square of 3x3 and went on to the 9x9. The 3x3 gives 9! and the 9x9 gives 9!*8!*7!*6!*5!*4!*3!*2!*1!, or 1,834,933,472,251,084,800,000 (1.83 e 21) different possibilities. (Nick raised the objection that in our calculation we did not account for the 3x3 rule (A number can only appear once in a 3x3 square) but after a little thought we think that the 3x3 rule is automatically satisfied if the column and row rules are satisfied (which we based our calculations on). So for calculation's sake we will make that assumption until further outcome of that debate.) Now that is just the number of ways a single puzzle can be filled. Several different puzzles can be made from the same arrangement of numbers by excluding different numbers.

During my public speaking class I found that the easy Sudoku in the paper has 40 numbers of the 81 possible, the moderate has 36 and the hard has 32. From there it starts getting messy. If we just consider the easy puzzles that will give us 3.987 x 10^44 different puzzles ( that comes from 81!/((81-40)!40!)*1.83 e 21). I could give exact numbers here but at the moment I'm thinking "what's the point". So the lesson here is, they have plenty of possible Sudoku puzzles to make, so they will most likely not run out of them.