Minimal Kropki Sudoku

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Minimal Kropki Sudoku

Postby jovi_al01 » Sat Aug 07, 2021 2:35 am

hello friends! i found something a few months ago while playing with a brute force solver and wanted to share here.
this is a unique kropki sudoku with only one clue.

kropki sudoku rules:
normal sudoku rules apply. cells separated by a white dot are consecutive. cells separated by a black dot have a ratio of 2:1. all dots have been given (i.e. two cells not separated by a white or black dot cannot be consecutive or have a ratio of 2:1).

the puzzle:
Code: Select all
.-------.-------.-------.
| . . . | . . . | . . . |
| . . A | B . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
.-------.-------.-------.


placing either a black or a white dot between A and B will result in a unique solution. fascinating, i think!

EDIT:: as far as i am aware, there is no logical approach to this puzzle.
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Re: Minimal Kropki Sudoku

Postby Mathimagics » Sat Aug 07, 2021 9:24 am

The solution is indeed unique, and the same for both puzzles (white-dot, black-dot). It has A=2, B=1:

Code: Select all
 +-------+-------+-------+
 | 4 1 8 | 5 9 6 | 2 7 3 |
 | 7 5 2 | 1 3 8 | 6 4 9 |
 | 3 9 6 | 4 7 2 | 8 1 5 |
 +-------+-------+-------+
 | 9 6 4 | 7 2 5 | 1 3 8 |
 | 5 2 7 | 3 8 1 | 4 9 6 |
 | 1 8 3 | 9 6 4 | 7 5 2 |
 +-------+-------+-------+
 | 8 3 1 | 6 4 9 | 5 2 7 |
 | 2 7 5 | 8 1 3 | 9 6 4 |
 | 6 4 9 | 2 5 7 | 3 8 1 |
 +-------+-------+-------+

I agree, fascinating! 8-)

I don't actually have a Kropki solver - I adapted (ie. hacked) my forbidden-pair solver to check these results ...
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Re: Minimal Kropki Sudoku

Postby jovi_al01 » Sat Aug 07, 2021 6:48 pm

wonderful!
i wonder if there is some potential "meta-knowledge" so-to-speak that might help one solve a kropki sudoku with very few clues, such as this.
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Re: Minimal Kropki Sudoku

Postby Mathimagics » Sat Aug 07, 2021 7:43 pm

I think that our experience with variants generally is that, the less clues a puzzle has, the less likely that it would be suitable for human solvers.

In extreme cases, like this one, it's very unlikely ...
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Re: Minimal Kropki Sudoku

Postby jovi_al01 » Sun Aug 08, 2021 3:00 am

certainly. non-consecutive is an extremely powerful constraint already, but coupled with the anti-ratio... i shouldn't be surprised that this exists, yet i am!
still, in "dotless" kropki puzzles (say some digits were given) there must be at least some information about all possible grids that are non-consecutive and anti-ratio that could aid in solving (roping, forced adjacencies, etc).
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Re: Minimal Kropki Sudoku

Postby Mathimagics » Sun Aug 08, 2021 8:11 am

By my reckoning, there are only 8448 "dotless kropki" grids, ie grids which have the NC property and also exclude the pairs {24, 36, 48}. So, excluding rotations/reflections, there are only 1056 ED grids.

There appear to be no "1C" dotless puzzles (dotless + one given). There are 2C puzzles, of course. Intriguingly, all of the 2C puzzles involve adjacent cells with givens {12}, {24}, or {36}.
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Re: Minimal Kropki Sudoku

Postby jovi_al01 » Tue Aug 10, 2021 3:23 am

that is wild! thank you for looking into it :)
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