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

(baldino.dev)

22 points by Fran3141 day ago

4 comments

Fran3141 day ago
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!
- redfast002 hours ago
 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_wrong1 hour ago
 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<pre><code> 1234 3412 2341 4123 </code></pre> If you rotate it counterclockwise you have<pre><code> 4213 3142 2431 1324 </code></pre> And if you normalize it, replacing the first row with "1234" (brilliant idea from the post), you get<pre><code> 1234 4312 2143 3421 </code></pre> 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.
gilleain1 hour ago
There is a way of understanding all this through group theory. eg:<a href="https://arxiv.org/html/2607.20669" rel="nofollow">https://arxiv.org/html/2607.20669</a>'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.
- qsort1 hour ago
 Can't you just describe the group that defines those transformations and use Burnside?
 - gilleain38 minutes ago
 Sadly my group theory knowledge is rudimentary, and rusty. Probably you could? No idea.
irusik1 day ago
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.
- Fran3141 day ago
  I'm glad this inspired something in you!
cole_ilands1 day ago
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