Wednesday 23 November 2011

Slitherlink Pattern Guide

This is an introduction to number patterns in Slitherlink. It's not a complete guide to how to solve these puzzles. It's about patterns I think are handy to recognise as it makes solving these puzzles easier. There's a few patterns that are very common, which most people who solve these puzzles are aware of. This is because they show up in about every other puzzle. I like to use the more unfamiliar patterns in my puzzles as it gives people new things to learn. In my opinion it doesn't actually make the puzzles more difficult, just takes more effort to solve as you have to discover the patterns first. If you already know them, they're not actually harder.

Slitherlink Patterns

I've divided them in a few categories. There's open space patterns, edge/cross patterns and corner patterns. The open space patterns can appear anywhere in the grid and even apply to toroidal variants. Edge/cross patterns need to be at an edge or already need a cross before being applicable. These could thus also be found in a toroidal variant. The corner patterns really only apply when in a corner of a grid. They will hardly ever apply anywhere else in the grid.
A lot of these patterns can have extensions and variants. I will show where these apply. There's a general rule for most of these extensions and variants, which I will also point out with these examples.
The pictures will show the patterns in black writing, with the deducable lines and crosses in red next to it. It's always important to also keep track of the crosses in Slitherlink puzzles, especially when dealing with mine. I like to build puzzles around the available crosses more than what's already drawn in lines. If you keep track of the crosses, it will be much easier to see the next steps than when you don't.

Open space patterns

The first patterns are probably familiar to many people. These are namely the orthogonally adjacent 3s and the diagonally adjacent 3s. I'm adding these two patterns to be thorough but also to show the extensions and variants.
Orthogonally Adjacent 3s

This is probably one of the first patterns people learn. It appears in many puzzles as it gives a lot of information. This pattern has no extensions, but it has a small variant. It usually appears as a 3-2-0 in line, but it only really requires a cross on the 2.

This pattern gives a few less lines, but still the same crosses. As you can see it looks very similar, and still can give some nice things to build a puzzle on with further.

Diagonally Adjacent 3s

This is another familiar pattern to most. Also seen in many puzzles. It's nice to be able to put in 2 numbers and be able to force all these lines and crosses. This pattern can be easily extended by placing a 2 in between.

This extension is probably also know to most people. It's at least mentioned most times by people explaining this pattern. It doesn't matter how many 2s are in between the 3s, it will always apply.
Besides an extension, there is also a variant on this pattern. It works similar to the variant shown above. Namely by replacing one of the 3s by a 2 with a cross.

This variant I would usually put in by putting a 0 next to the 2. This way it's a good opening pattern.This variant could also be applied on the extensions. I will leave that picture out as I think you get the idea now.

1-3-1 Pattern

I always like putting in this pattern. With this you can create corners in the middle of the grid. It doesn't give any lines, which might get people who don't tend to keep track of crosses properly stuck in the puzzle. These corners can be used to build the puzzle on further.

This pattern can be easily extended. Most extensions work in the same way, by moving a 3 further away by placing it diagonally from a 2. It can be extended with multiple 2s as well. I used this pattern to open up this puzzle. That puzzle is a very hard one. But it does show well how you can exploit this pattern.

3-1-1-3 Pattern

This pattern is based around two orthogonally adjacent 1s, with a 3 on both long sides. The 3s can be in any of the two squares on either side and the pattern still works. So you could also have both 3s on the left or right and you'd still get the same lines and crosses. This one I don't use often, but I still like it. I especially like the extended versions. They can work with other numbers close by very well. The extended version is another diagonal 2 construction.

What I like about this extension is that there is a line leading to a square, which doesn't have any numbers adjacent to it. So that would be a good place to place a 3. This pattern can again be extended and can even be placed on both sides. The first time I spotted this pattern I was suprised there was something to get there, which is probably why i like it.

Edge/Cross Patterns

These patterns are usually found on the edge of a grid. I like to incorporate them in the middle of the grid as well. This is possible as they usually only need a single cross to warrant the pattern correct. When on the edge of the grid, this cross is implied as the loop can't go through the edge of the grid.

1-3 Pattern

This is the most familiar pattern of this variety. In the easier puzzles you will usually see it as a 3-1-3 on an edge, which is basically this pattern twice and gives a big chunk of the loop. The single cross next to a dot between the 1 and 3, already holds the elimination true. This is why you can also regularly spot it in the middle of a grid. This pattern can also be extended like almost any pattern with a 3.

Again, this could be extended with as many diagonal 2's as you want. Opposed to the unextended pattern, I think this one will go unnoticed to more solvers, even though it's really the same pattern. Of course you can still make these deductions logically as well. It's just easier to know what to put down without having to think about it.

2-1-3 Pattern

This pattern could at first glance be misinterpretted as an extension of the previous pattern, but the 3 is in the wrong direction. It doesn't lead to the same deductions. Instead it gives you some lines. It's not a pattern I had readily stored in my head. But I noticed it in my puzzles and thought it would be handy to know for everyone.

This pattern also has an diagonal 2 extension. It's always smart to check whenever a 3 is diagonally from a 2, if there isn't an extended pattern hidden. It won't be necessary in many puzzles to use these extended patterns. It will give you a possible shortcut in the puzzle though. This can't be a bad thing.

Corner Patterns

 I didn't manage to include very many edge patterns. I just don't want to include too many obscure ones. They are really only handy for really hard computer-generated puzzles. Corner patterns are more common though. The first two I find very standard. They don't give that much, but they are patterns I feel you should mark whenever you see them.

1-1 Corner

I always place these crosses whenever this pattern shows up. It's easily exploited by placing a 2 in the right bottom square and creating a corner 2.

1-3 Corner

This is really the same as the above one, except you get a line and a cross. I never miss out placing this pattern because it's so easily spotted.

2-3 Corner

This one is most likely also known by people, but maybe not always instantly put down. It is another one of those corner deductions that doesn't give very much directly, but it will help when you need it if you have already put it down. This pattern has a small extension that I usually put down as the added 1 will give more to work with in the long run of the design.

It's only one extra line. But the way this extra line is forced is something you should really remember for Slitherlink puzzles. If that line wasn't true, you'd force a second loop.

2-3-2 Corner

This pattern is really a favourite of mine. You get four lines, two of them pointing away from the pattern. This gives easy ways to continue the design of the puzzle.

Odd 2's Corner

This deduction is familiar to everyone. What most likely isn't known to everyone is that this deduction holds true for any odd amount of 2's in a diagonal line.

This is an example with three 2's. If these lines weren't true you'd lead three ends into the corner and you wouldn't be able to make a loop. This will also work for any other odd number of 2's. It can be a fruitful pattern if used correctly. Check  this puzzle for example. Using this pattern as the opening move makes the whole puzzle work very nice.

3-2 Corner Pattern

This is probably the most important pattern to remember though when it comes to corner patterns. This is because this one can easily happen in the middle of the grid. You don't need the edges of the grid to make this pattern true. You only need the 2 to be in a corner for it. As shown with previous patterns, it's relatively easy to force a few crosses, which can put a 2 in a corner. In this puzzle you can see many of the center 2's being forced into a corner by crosses and will also see this pattern pop up.

As you can see this pattern can be extended easily, as a 2 diagonally from a corner 2 will also be in a corner. I want to point out to not forget the crosses for this pattern as especially the cross between the 2 and 3 can sometimes be forgotten.

Miscellaneous Pattern

I'm adding this section, just to point out an observation I used in this one to really break open the puzzle. It's something most people will be familiar with, I just wanted to show that it can be extended as well.

U-turn-3 Pattern

This observation won't be lost on anyone. If you were to draw lines where the crosses are, you'd also have to make a U-turn around the 3 and create a mini loop. This isn't allowed as you can only create a single loop. This way this pattern looks very obvious. But you can hide it a little by extending it.

This extension is based on the fact that a square diagonally from a 2 in a corner is also a corner. We used this reasoning in the previous pattern as well. It isn't as obvious that drawing any of these lines will result in the same U-turn, but it still holds true.

Puzzle #51

As I had mentioned in puzzle #51, I wrote that puzzle to apply many of these patterns. So I figured I's show you all what I had in mind as a starting point when designing this puzzle.

I've drawn in all deductions to be drawn from the patterns I had mentioned. Each pattern has a seperate colour. I've added some crosses for the 0's, which sometimes apply to more than one pattern. I just wanted to make clear where the necessary crosses came from, to make the pattern true.
You can see here how the 1-3-1 Pattern initiates in the 3-2 Corner Pattern. So even though it doesn't give any lines, it can still be useful to make progress in the puzzle.
When you solve this puzzle with the now given deductions, you'll see how much easier it goes. I know not all patterns are necessarily needed, but you can see how much I put in to start you off.

I hope this was all clear and helpful. I think this has given people some good tricks to either solve or create their next Slitherlink. I don't think you'll need all of these in most puzzles, but if you run into more challenging puzzles some of these patterns might get you easier through some sticky points.
I'm planning on writing a follow up post about Slitherlink as well. This is to address some advanced techniques that I think are useful as well. Many of these tricks not only apply to Slitherlink, but more to loop puzzles in general.

If you have any questions or comments, feel free to leave them on this post. 


  1. There were definitely a lot of things here I didn't notice as patterns before, thanks! Puzzle #51 solved a lot more easily from this starting point, though even knowing these patterns now I doubt I would have spotted that green U-turn.

  2. Thank you so much for this gem of a blog post. Sadly, I did not appreciate a few of these already, and will incorporate going forward.


  3. You're welcome. It was fun to write, although it took far longer than I had expected.

    I think the green U-turn is the hardest to see. I also don't really think it's a pattern you will spot at the start of the puzzle. But it is the definite breaking point for the left side of the puzzle. When I watched people solving this puzzle, seeing that deduction really broke open a lot of the puzzle.

  4. Thanks for these patterns. some of them were new for me.

  5. I thought i was quite good at this... apparently not. Great tips!

  6. thanks for codifying this. love it.

  7. Extended Variant 3-3 side by side

    ...2 .......2
    |3|3| -> |3|3|

    ...2 .........2
    |3|3|3| -> |3|3|3|

  8. This article is fantastic. Many of these advanced patterns I've not seen on any other site.

  9. Such a great post, you have me obsessed with Slitherlink. You'd mentioned putting together a post of advanced patterns, that would be fantastic! Any timeframe?

  10. does the 3 2 pattern have to be surrounded by no other numbers to work?

  11. This is such a good article that I keep on coming here again and again. Thanks Bram for these patterns.