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Some background first. The gimmick for this puzzle was inspired by a recent puzzle in a mathematics discord. As part of solving that problem I learned how to use a SAT Solver to generate cases.

During the process of solving that puzzle, I came up with the idea for a sudoku variant using some of the rules of that puzzle. Unfortunately, I have never actually designed a sudoku puzzle before, so I adapted my SAT Solver to help me through it.

I am reasonably sure that this puzzle has a unique solution. However, I cannot promise that the solution is reasonable to find. Examples of grids that satisfy rules 1-6 are actually quite rare and are not at all easy to find even with computer assistance. I have attempted to shape at least some of the green cages in such a way that it reveals information, but I have not managed to crack it myself by hand (though I have tried, I am by no means a sudoku expert, so this is not a strong proof of human impossibility).

With those caveats out of the way, if you're looking for what I suspect might be a monstrously difficult, but perhaps robotically unfun, puzzle, read on.

Rules:

  1. Each cell gets a digit from 1-9 inclusive.
  2. Each row and column contains all the digits from 1-9
  3. A given digit may be unique in both its row and column, or it may repeat any number of times within its row OR within its column, but it may not repeat in both its row AND its column.
  4. There are an equal number of instances of each digit.
  5. Within each black 3x3 square, there are exactly 4 distinct digits, and each digit appears at least twice.
  6. Within each red 4x4 square, there are exactly 9 distinct digits.
  7. Each green cage has the same sum.

sudoku variant image

And here is a version that a friend made which has the green cages colored in for visual clarity.

sudoku variant image with colored cages

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  • $\begingroup$ Additionally, if someone would like examples of such grids, I would be happy to provide a few of them. I don't believe I did the idea justice, but if someone better at making sudokus than I am wants to take a crack at it, I would be honored to help. $\endgroup$ Commented Oct 14 at 20:38
  • $\begingroup$ Cool idea! Does the [computer-puzzle] tag mean that you expect this puzzle to be solved via computer program, or are you looking for a human-understandable solve path? Note that the best way to create a puzzle satisfying to solve by hand is to, well, create it by hand. $\endgroup$ Commented Oct 14 at 21:19
  • $\begingroup$ The computer puzzle tag means I used a computer to create it, and I don't know whether or not there is a human understandable path to solve it. That being said, I am very interested in knowing if there is $\endgroup$ Commented Oct 14 at 21:32
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    $\begingroup$ No, the green cages have as few as 6 and as many as 11 $\endgroup$ Commented Oct 14 at 22:42
  • $\begingroup$ rot13 : Fhz bs qvtvgf vf sbegl-svir, fb gur fhz bire nyy qvtvgf va gur chmmyr vf sbhe-uhaqerq-svir. Gurer ner fvkgrra pntrf juvpu fubhyq nyy unir gur fnzr fhz. Ubjrire fvkgrra qbrfa'g qvivqr sbhe-uhaqerq-svir? $\endgroup$ Commented Oct 15 at 6:43

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