'Counting, Symmetries and Equivalence Classes of Sudoku Grids' where an 'equivalence class' is a set of structures (such as filled Sudoku grids) that are all equivalent under some relation.
In the last few days I went down the rabbithole of 4x4 sudokus, and found out that (up to permutations) there are only 12 possible solutions.
I decide to write up the small research I did as well as some fun findings discovered along the way in a blog post.
Also, yes this is extremely pointless and silly, and the math involved is not incredibly high level, but I still think it's an enjoyable bit of recreational math worth your time!
Did you check for rotational and mirror symmetry? As in, are the 12 unique sudokus just the same one, rotated in 4 ways, and mirrored along the x or y axis?
I was wondering this myself so I wrote a script that shows that there are duplicated puzzles if you count a rotated Sudoku as being equivalent. Here's one example:
Puzzle 2 is
1234
3412
2341
4123
If you rotate it counterclockwise you have
4213
3142
2431
1324
And if you normalize it, replacing the first row with "1234" (brilliant idea from the post), you get
1234
4312
2143
3421
Which is listed as puzzle 7. A quick check gives me that puzzles 1, 2, 3, 5, 11 and 12 would be unique under rotation, but I wrote that check in five minutes so there must be bugs somewhere. Also, I performed no check for mirror symmetry whatsoever (update: I did a quick one, same result).
I don't want to miss my chance to say that the post is brilliant and that it convinced me to leave what I was doing to check for rotations. I was nerd sniped in the best way and I take my hat off for the OP.
I love articles where a seemingly simple puzzle turns out to be much more interesting when you look at it from the perspective of math and code. It’s especially interesting to learn that there aren’t actually that many possible 4×4 Sudoku grids. I also enjoy working with puzzles and creating my own crosswords in SuperColoring. Articles like this make me want to try creating a more unusual crossword and see how much harder it would be to solve.
There is a way of understanding all this through group theory. eg:
https://arxiv.org/html/2607.20669
'Counting, Symmetries and Equivalence Classes of Sudoku Grids' where an 'equivalence class' is a set of structures (such as filled Sudoku grids) that are all equivalent under some relation.
In the last few days I went down the rabbithole of 4x4 sudokus, and found out that (up to permutations) there are only 12 possible solutions.
I decide to write up the small research I did as well as some fun findings discovered along the way in a blog post.
Also, yes this is extremely pointless and silly, and the math involved is not incredibly high level, but I still think it's an enjoyable bit of recreational math worth your time!
Did you check for rotational and mirror symmetry? As in, are the 12 unique sudokus just the same one, rotated in 4 ways, and mirrored along the x or y axis?
I was wondering this myself so I wrote a script that shows that there are duplicated puzzles if you count a rotated Sudoku as being equivalent. Here's one example:
Puzzle 2 is
If you rotate it counterclockwise you have And if you normalize it, replacing the first row with "1234" (brilliant idea from the post), you get Which is listed as puzzle 7. A quick check gives me that puzzles 1, 2, 3, 5, 11 and 12 would be unique under rotation, but I wrote that check in five minutes so there must be bugs somewhere. Also, I performed no check for mirror symmetry whatsoever (update: I did a quick one, same result).I don't want to miss my chance to say that the post is brilliant and that it convinced me to leave what I was doing to check for rotations. I was nerd sniped in the best way and I take my hat off for the OP.
I love articles where a seemingly simple puzzle turns out to be much more interesting when you look at it from the perspective of math and code. It’s especially interesting to learn that there aren’t actually that many possible 4×4 Sudoku grids. I also enjoy working with puzzles and creating my own crosswords in SuperColoring. Articles like this make me want to try creating a more unusual crossword and see how much harder it would be to solve.
I'm glad this inspired something in you!