User talk:Cimagna

I recently read an article that enumerates all of the solutions to a sudoku game. I have a simpler method of enumerating all solutions; correct me if I'm wong:

Let's start in row #1. There are 9! distinct arrangements of the digits 1,2,3, ... ,8,9, i.e., 9! distinct arrangements in row 31.

For each such arrangement, let's enumerate all the valid arrangments for row #2, as follows: The left-most square of row #2 can contain eight possible digits from the set { 1, ...,9 } since one of those digits is already assigned in row #1 square #1. The next square can can be assigned 7 of the remaining 8 digits since one of the remaining digits is already assigned to row #1 square #2. Continuing this line of reasoning until we reach square # 9 in row #2 and the last digit must be valid in this square since the other eight digits have been successfully assigned in row #2. How many valid arrangements of row #2 have we generated for each arrangement in row #1? It is 8x7x6x5x4x3x2x1 = 8! arrangements. Hence, so far we have generated 9! x 8! arrangements of row#1 and row #2 that satist the rules of sudoku. Using the same reasoning on row #3, for each arrangement of row#1/row#2 pair, we will generate exactly 7! arrangements of row #3. If we continue this process down to row #9 we will have generated 9!x8!x7!x6!x5!x4!x3!x2!x1! = 1,834,933,472,251,084,800,00 solutions to sudoku. This seemed too simple judging from all the work that's been done to find this number. Have I missed something?