> Using combinatorics, we can take any one Sudoku grid and, with various simple tricks, create enough unique grids for you to do one each day for the next century. Simply by transposing and rotating the grid or interchanging columns and rows we get exponentially more unique puzzles.
You can sort the three columns of boxes in six ways, same for the rows of boxes. Within each column of boxes there are three columns that can be sorted in six ways - same again for the rows. That gives us (6 × 6) × (6 × 6 × 6) × (6 × 6 × 6) = 1 679 616 variations. You can also turn the grid 90 degrees, 1 679 616 × 2 = 3 359 232.
Then you can shuffle the order of the nine digits, since they don’t have any mathematical meaning, but are just arbitrary symbols. That’s another 9! = 362 880.
3 359 232 × 362 880 = 1 218 998 108 160
That’s going to keep you busy for more than the next century. In fact, even if you solve one per minute, you’ll be busy for 2.3 million years solving permutations of a single sudoku.
Funny that you bring this up. These puzzles would not be considered distinct within group theory. A simple application of Burnside’s lemma would allow a person to recover the true number of distinct puzzles from this figure.
A person playing the same puzzle every day that’s just been rotated or had the symbols permuted is going to quickly get bored. While the puzzle looks superficially different the graph structure and the logical steps in the solution will be identical. So doing this kind of combinatorial shenanigans is “cheating” and should be considered dubious, if not fraudulent, within the sudoku community.
I always thought it would be a fun prank to create a book of sudokus that were all permutations of the same puzzle. See how long it would take for someone to notice that they are solving the same puzzle over and over again.
No, that could definitely happen. Not sure how to estimate how many duplicates there would be. It would have to depend on the specific puzzle you started with.
You can sort the three columns of boxes in six ways, same for the rows of boxes. Within each column of boxes there are three columns that can be sorted in six ways - same again for the rows. That gives us (6 × 6) × (6 × 6 × 6) × (6 × 6 × 6) = 1 679 616 variations. You can also turn the grid 90 degrees, 1 679 616 × 2 = 3 359 232.
Then you can shuffle the order of the nine digits, since they don’t have any mathematical meaning, but are just arbitrary symbols. That’s another 9! = 362 880.
3 359 232 × 362 880 = 1 218 998 108 160
That’s going to keep you busy for more than the next century. In fact, even if you solve one per minute, you’ll be busy for 2.3 million years solving permutations of a single sudoku.