You are given a 9x9 grid of numbers such that:
a. each row,
c. each of the nine 3x3 grids
has each number 1 through 9 exactly once.
Lets call it Sudoku puzzle's solution or SPS.
A grid with condition c. removed is called a Latin Square, LS.
1. (d2) how many 9x9 LS's are there?
2. (d4) And how many 9x9 SPS's ?
For the LS: start with an empty 9x9 grid. Put the 1's in, one in each row, never one below another. At the 9th row, there is only one available spot. There are 9! ways to do this. Next, put in the 2's. Since each successive row has one less available spot, there are 8! ways to do this. By the 8th and 9th row you will find only one spot on the 8th row that allows a legal placement of a 2 on the 9th as well. Likewise, with the 3's. At row 7 there will be only one allowed set of placements for rows 7, 8, and 9. And, so forth. (I wish I could prove why this is so. Anyway, it makes the answer simple).
For the 9x9 LS, N = 9! 8! 7! 6! 5! 4! 3! 2! ~ 1.8 10^21
Edited on July 14, 2018, 4:39 am