# What are the rules to my Sudoku?

I made a Sudoku variant puzzle pictured here:

Your puzzle is to figure out what the rules for the puzzle I created are.

The puzzle has a single unique solution so if your rule-set allows more than one solution then it's wrong.

The rules I have in mind are not very complex and are easy to understand, so you should avoid complex, esoteric solutions.

• Did you, by any chance, use a satisfiability solver (instead of a manual test solve, or even a deduction technique based sudoku solver) to guarantee the uniqueness? I think I got your rule set (I may be wrong, of course), but proving uniqueness seems to require bifurcation, also known as brute forcing every possibility. (I may just be bad at solving sudoku, of course)
– Bass
Oct 2, 2021 at 9:21
• @Bass I didn't know of anything off the shelf that would work for this and I didn't feel like writing my own. So I just used manual checking. Oct 2, 2021 at 10:14
• I'd recommend maybe giving a hint now it's been a couple days with no progress :) Oct 4, 2021 at 0:11
• @BeastlyGerbil That seems like a good idea, but I don't know how to give a hint that makes the puzzle easier without completely trivializing it. Oct 4, 2021 at 13:28
• If you can just think of something vague perhaps, or even a hint that would only make sense to someone who’s thinking of the correct rules, that could work. Hints don’t always have to tell you the next step! But I can see how it’ll be a bit trickier with reverse puzzling Oct 4, 2021 at 13:44

I have a potential rule set, that almost allows for manual-solving using normal sudoku-like deductions:

Each 4x4 region must contain every number from 1-16 exactly once.
Mirrored pairs of rows/columns (R1+R8, R2+R7 etc.) must also contain every number from 1-16 exactly once.

Applying those rules gets me to the following position.

10 8 14 3 | 16 6 7/11 9​
 2 13 7 6 | 5 1 15 4/8
12 1 15 11 | 4/7 2 10 14
16 5 4 9 | 13 8/11 12 3
------------------+------------------
15 6 8/11 14 | 1 10 2 7
 5 3 13 4/7 | 8 9 16 6
4/8 9 16 10 | 12 3 14 11
 1 7/11 12 2 | 15 5 4 13

Which has two valid solutions:

10 8 14 3 | 16 6 7 9​
 2 13 7 6 | 5 1 15 8
12 1 15 11 | 4 2 10 14
16 5 4 9 | 13 11 12 3
---------------+---------------
15 6 8 14 | 1 10 2 7
 5 3 13 7 | 8 9 16 6
 4 9 16 10 | 12 3 14 11
 1 11 12 2 | 15 5 4 13

and

10 8 14 3 | 16 6 11 9​
 2 13 7 6 | 5 1 15 4
12 1 15 11 | 7 2 10 14
16 5 4 9 | 13 8 12 3
---------------+---------------
15 6 11 14 | 1 10 2 7
 5 3 13 4 | 8 9 16 6
 8 9 16 10 | 12 3 14 11
 1 7 12 2 | 15 5 4 13

So either the I have the rules wrong, or I've made a mistake during solving.

• Yeah, so it's the rules that are wrong, because that's my exact rule set. Sorry for that I had myself and a second party try it to make sure the solution was unique. We must have made the same error. Oct 7, 2021 at 12:24