How Are Sudoku Permutations Calculated?

[JWilliams]

Hello all,

I've recently developed an obsession with the game Sudoku. As I seek to lern more about the puzzle game, my ramblings around the internet led me to the wikipedia page describing the mathematics of Sudoku. The portion of interest to me right now is: http://en.wikipedia.org/wiki/Mathematics_of_Sudoku#Band1_permutation_details"

I understand how they are defining the different interactions between the different cases, however, what is puzzling me is how they are arriving at 3³x6³ as the numerical permutations contributed by the triplets in case 1 and 2. My knowledge of permutations is somewhat limited however I understand why the first box contributes 9! combinations and the final box contributes 3!³ combinations.

Any insight would be greatly appreciated.

 

[mmwave]

Hello there,

It's great to hear that you have developed an interest in Sudoku and are seeking to learn more about it. I can provide some insight into the mathematics behind Sudoku and how the numerical permutations are calculated in cases 1 and 2.

In Sudoku, the goal is to fill a 9x9 grid with numbers from 1 to 9, with each row, column, and 3x3 subgrid containing all the numbers exactly once. This means that each number can only appear once in each row, column, and subgrid.

Now, let's focus on case 1 and 2 mentioned in the Wikipedia page. In these cases, we are looking at the interactions between the numbers in three different rows or columns within a single subgrid. This means that the numbers in these three rows or columns must be distinct, and they can appear in any order.

In case 1, we have three rows or columns that contribute to the permutations. This means that for the first row or column, we have 9 options, for the second row or column, we have 8 options, and for the third row or column, we have 7 options. This gives us a total of 9x8x7 = 504 permutations for case 1.

In case 2, we also have three rows or columns that contribute to the permutations. However, in this case, we have two distinct numbers that can appear in any order in the first row or column, and one number that appears in the second and third rows or columns. This means that for the first row or column, we have 9x8 = 72 options, and for the second and third rows or columns, we have 7 options each. This gives us a total of 72x7x7 = 3528 permutations for case 2.

Therefore, the total number of permutations contributed by the triplets in cases 1 and 2 is 504 + 3528 = 4032. This is equivalent to 33x63, as mentioned in the Wikipedia page.

I hope this explanation helps clear up any confusion you had about the numerical permutations in Sudoku. Keep exploring and learning about this fascinating puzzle game!