How Are Sudoku Permutations Calculated?

In summary, the mathematics behind Sudoku involves calculating numerical permutations in different cases. In cases 1 and 2, we are looking at the interactions between numbers in three different rows or columns within a single subgrid. Based on the rules of Sudoku, the number of permutations in case 1 is 504 and in case 2 it is 3528, resulting in a total of 4032 permutations or 33x63. This is achieved by considering the distinct numbers and their possible arrangements in each row or column.
  • #1
JWilliams
1
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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.
 
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  • #2


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!
 

FAQ: How Are Sudoku Permutations Calculated?

What is a Sudoku puzzle?

A Sudoku puzzle is a logic-based number placement puzzle where the objective is to fill a 9x9 grid with digits so that each column, each row, and each of the nine 3x3 subgrids contains all of the digits from 1 to 9.

How many possible Sudoku puzzles are there?

There are 6,670,903,752,021,072,936,960 possible Sudoku puzzles, but only a small fraction of these are unique and solvable.

What is the minimum number of clues needed to solve a Sudoku puzzle?

The minimum number of clues needed to solve a Sudoku puzzle is 17. This has been proven by mathematicians, and any puzzle with fewer than 17 clues will have multiple solutions.

How do mathematicians create new Sudoku puzzles?

Mathematicians use various techniques and algorithms to generate new Sudoku puzzles. These techniques involve starting with a completed grid and then removing a certain number of clues while ensuring that the resulting puzzle is still solvable.

Is there a mathematical strategy for solving Sudoku puzzles?

Yes, there are many mathematical strategies and techniques for solving Sudoku puzzles. These include scanning, cross-hatching, and using logic and deduction to eliminate possibilities and fill in the grid. Some puzzles may require more advanced techniques such as X-Wings or Swordfish.

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