Considering Sudoku 2
Using the rules and goals of a Sudoku puzzle there are many thousands of possible arrangements of digits that would satisfy the conditions for a proper Sudoku solution.
Given a proper Sudoku solution. How many distinct (unique) arrangements of the digits in the grid could be presented by swapping entire rows or entire columns?
For example, swap all the digits between the top two rows, so that all of to the digits originally in the second row are moved up one cell into the top row and all the digits originally in the first row are move down one cell into the second row. That swap would produce a different arrangement in the grid, but would still be a legitimate correct solution.
The answer is 1679616.
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1 solution
As long as the digits within each horizontal row, vertical column and block remain unchanged the solution will be valid. Considering the rows. rows 1,2,3 The digits in any two of these three rows can swap positions in the grid the solution will hold.
3! =6 possible arrangements within each of the three groups
rows 1,2,3 can be swapped with rows 4,5,6 or rows 7,8,9 The digits in any of these groups can be swapped with the digests in any other group as long as the entire group is moved as a unit.
3! =6 possible arrangements of the three groups.
All of this is also true about the Columns as well as the rows.
(3!) ^ 8 = 1,679,616.
What is the proof that every one of the possible 1,679,616 arrangements of digits within the grid is in fact unique? Is there a possibility that there are duplicates?
Using a more symmetrical solution like this one as an example.
By swapping rows only, it is the equivalent to rearranging the row numbers into different permutations. All of the permutations of the row numbers are unique and therefore the grid layout will be unique. The same is true when rearranging only the columns. If there is duplication of the grid it must be a result of the changes made to the rows and the changes made to the columns combining in some way to reproduce a duplicate grid layout. Each time two rows are swapped there are 18 individual cells changed. Two rows have 9 changes made within them and all nine columns have two changes made within them. When manipulating rows an even number of rows are effected and an odd number of columns are effected with each change. When manipulating columns an odd number of rows (9) are effected and an even (2) number of columns are effected, with each change.
That is a convoluted explanation. I am actually hoping that someone night point out the flaw in my solution or post a more concise prof that the arrangements most be all different from each other.