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Please read and understand my first question before proceeding. This one should be worse.

pdf files I uploaded to Dropbox:

Slitherlink of Twos

Their solutions

These puzzles were generated by computer. A good solver has no problem solving them, so I can't just ask for people to find the solutions. That's why I'm just giving the solutions to you up front. Fair warning, these are super hard to solve by hand.

I did think of a different question I could ask, but we need to define some things first.

A Slitherlink of Twos is a typical, square cell, Slitherlink puzzle where each cell is either a "2" or left blank. It may have any grid size, even rectangular ones, just nothing weird like an "L" shape. It must have exactly one loop for a solution.

Let us also define backfill as the process of, having solved a Slitherlink puzzle, going back and filling in each blank cell with the appropriate number for the solution.

As we learned from the previous question, for a Slitherlink of Twos, we would expect that when we backfill a typical puzzle we would at some point be adding ones or threes.

Can you, when you backfill any Slitherlink of Twos, add a zero to a cell?

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  • $\begingroup$ Thanks to this question I know about slitherlink $\endgroup$ – Yout Ried Feb 6 at 19:32
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    $\begingroup$ @Yout Reid - It surprises me that Slitherlink isn't more popular, because I would place it well above games like Sudoku or Kakuro. It's funny that you learned about it through this question, because this is also the hardest Slitherlink question I could ask. You know... until I think of another one. $\endgroup$ – Dark Thunder Feb 7 at 10:18
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Well, look what I found. Another shot at a Slitherlink!

Answer:

Yes, there is (at least) one solution.

Explanation:

I believe this is the only solution that will work:

Solution

The real key here is to know some of the intricacies of Slitherlink:

1. If you have one or more zeroes along the edge of a puzzle, it would necessitate ones and/or threes in the puzzle elsewhere. This means that the zeroes have to be somewhere in the middle of the puzzle. Here's a couple of examples:

Large near-solution
Larger near-solution

2. Slitherlinks can only have one loop; you cannot have concentric loops as in the following:

"Solution" with concentric loops

With these things in mind, I do not see another way to have a Slitherlink puzzle with only zeroes and twos than the one that I found. I would be interested to see if someone's found another way!

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    $\begingroup$ Awesome! I never would have suspected that, and you answered another question I had which is whether or not a 10x10 could backfill a two. Without really knowing why other than intuition, even grid sizes are much harder and being both even and square is worse. My solution comes from the fact that almost all 5x5 Slitherlink of Twos (that I came across, anyway) have a zero in them. $\endgroup$ – Dark Thunder Feb 6 at 21:51

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