In Sudoku, there is an NxN matrix of integers from one to N. This has the property that no column or row has an integer repeated.
question: How many UNIQUE puzzles exist?
Some analysis:
The TOP row has N! has possible permutations.
For a 5X5 matrix, there are 44 second rows for each of the 120 permutations of the top row.
I did an exhaustive count of 5x5 puzzles, the total is 262180. This number is factorable, but the factors are not obvious. For each Row 2, you can either 12 or 13 row 3's, And for each row 3, you can get either 2 or 4 row 4's. Row 4 gives EXACTLY one choice for row 5.
It is tempting to say " It is a combination of ALL permutations of rows and columns," but this answer (120*120 = 14400 ) is not correct.
I am offering a cash prize for the correct formula. Maybe you know a math whiz that can provide it?