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.
'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.
That's interesting, less than I thought. I'm curious about this because I'm working on a sudoku variant that has two pieces of data in each cell - such as numbers and letters. I am keen to try out your process with that arrangement.
In addition to the (n^2)^2 sudokus, you can also make them with rectangular sub-blocks -- (n x m) ^ 2 sudokus -- such as a 6x6 grid with six (3x2) sub-blocks, or a 10x10 with 10 5x2 sub-blocks.
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!
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.
Can't you just describe the group that defines those transformations and use Burnside?
Sadly my group theory knowledge is rudimentary, and rusty. Probably you could? No idea.
This looks related to https://oeis.org/A107739.
If the 4x4 sudoku has 288 / 4! = 12 distinct solutions, then does the 9x9 sudoku have 6670903752021072936960 / 9! distinct solutions?
Or, for the unique solutions: https://oeis.org/A285180
Here is one possible way of generating all 12 solutions by 2x2x3 choices:
1. Fill the upper left box with 1-2-3-4
2. Choose where to put the 1 in the top right box (2 choices)
3. Choose where to put the 1 in the lower left box (2 choices)
4. Choose which digit to put diagonally opposite the 1 in the lower right box (3 choices)
Is there a nicer way which makes it obvious that there is exactly one solution for each choice in the last step?
> Choose which digit to put diagonally opposite the 1 in the lower right box (3 choices)
There are only two choices there. You cannot put a 1, nor the digit (3 or 4) that’s in the top the column where you try to put the number.
> Is there a nicer way which makes it obvious that there is exactly one solution for each choice in the last step?
There isn’t. You may end up with a degree of freedom after step 4
leads to which allows for 2 solutions:My procedure starts with filling the top-left box, not the top row. So it looks something like this after the third step:
From here x can be any of [2,3,4], and each yields exactly one solution!That's interesting, less than I thought. I'm curious about this because I'm working on a sudoku variant that has two pieces of data in each cell - such as numbers and letters. I am keen to try out your process with that arrangement.
In addition to the (n^2)^2 sudokus, you can also make them with rectangular sub-blocks -- (n x m) ^ 2 sudokus -- such as a 6x6 grid with six (3x2) sub-blocks, or a 10x10 with 10 5x2 sub-blocks.
I’ll plug a 4x4 variant sudoku generator that Claude put together a few months ago.
https://yakymp.github.io/sudoku4x4/
[dead]