Yet another Hacker News

There are only twelve 4x4 sudokus (and a cool trick for finding minimal subsets)

31 points Fran314 1 day ago 12 comments

Fran314 1 day ago | parent

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!

redfast00 3 hours ago | parent

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?

probably_wrong 3 hours ago | parent

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.

irusik 1 day ago | parent

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.

Fran314 1 day ago | parent

I'm glad this inspired something in you!

gilleain 3 hours ago | parent

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.

qsort 3 hours ago | parent

Can't you just describe the group that defines those transformations and use Burnside?

gilleain 2 hours ago | parent

Sadly my group theory knowledge is rudimentary, and rusty. Probably you could? No idea.

_usefulcat 1 hour ago | parent

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.

cochleari_major 57 minutes ago | parent

I’ll plug a 4x4 variant sudoku generator that Claude put together a few months ago.

https://yakymp.github.io/sudoku4x4/

drmajormccheese 20 minutes ago | parent

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?

Polizeiposaune 10 minutes ago | parent

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.