OddThinking

Introducing Slitherlinks

Slitherlinks are a type of puzzle I have been enjoying for a while.

Briefly, the aim is to draw a line the edges marked on a square grid, such that two constraints are met.

• The line forms exactly one loop.
• Some of the squares are marked with a number from 0 to 3. Those squares should be bounded by that many line segments. For example, a 3 should have exactly three of the four sides inked in, and one side left blank.

Examples of the puzzle (and better descriptions of the rules) are available from Conceptis Puzzles, Slither Link Cosmos and Nikoli.

The rest of this article includes some Slitherlinks tips for players who understand the basics and want to tackle the medium and hard examples.

Solving Slitherlinks

The puzzle requires a certain amount of trial and error, and a large amount of logic to solve.

Normally, there are some obvious starting points, like eliminating the borders of the 0 clues. Then you need to start testing some possibilities.

After a while, you will start to recognise some patterns that allow you to shortcut the trail and error process. The more patterns you recognise the faster you can find the Slitherlinks solutions.

Below I describe the most common patterns that I use to solve the puzzle. There are certain to be many more, but these are the ones I find to be most useful.

Notation of Slitherlinks Examples

In the examples below, the following colour scheme is used:

• Blue lines represent edges that have been inked in.
• Faint yellow lines represent edges that have been blanked out. (Other web-sites indicate this with a red cross.)
• Grey lines represent edges that are still unknown. (Other web-sites leave these white, but I find this clearer.)
• Black numbers represent clues that haven’t been solved.
• Faint yellow numbers represent clues that have all four edges solved, and therefore have become irrelevant to the final solution.

The examples below are artificial and only show the detail necessary. In the real puzzles, these patterns would be surrounded with the clutter of more clues and other partially solved edges.

In each pair of pictures, the left is the example of the pattern to find, and the right is the example of the conclusion that can be drawn.

Finally, only one or two examples of the patterns are provided. Be sure to look for rotated or flipped versions of these patterns.

Patterns

3 in a corner

Pattern Example	Conclusion

3 in a virtual corner (similar to above)

Pattern Example	Conclusion

2 in a corner

Pattern Example	Conclusion

2 in a virtual corner (similar to above)

Pattern Example	Conclusion

1 in a corner

Pattern Example	Conclusion

1 in a virtual corner (similar to above)

Pattern Example	Conclusion

Two 3s adjacent

Pattern Example	Conclusion

Two 3s diagonal

Pattern Example	Conclusion

1 on edge

Pattern Example	Conclusion

1 on virtual edge (similar to above)

Pattern Example	Conclusion

3 adjacent to 1 on edge

Pattern Example	Conclusion

Advanced Techniques

Even Crossings of a Jordan Curve

Here’s an interesting technique. Draw a loop around any part of the grid. The line you draw shouldn’t cross itself. I am going to call the loop a “Jordan curve” here (for historical reasons.)

The total number of inked-in grid lines touching your Jordan curve has to be an even number.

(If there are numbered clue squares on both the inside and the outside of the Jordan curve, you can go further and say that there must be at least two lines crossing.)

Here’s an example of that in action.

Even Crossings of a Jordan Curve

Pattern Example	Super-imposed Jordan Curve	Conclusion

Metagaming

Here’s another advanced technique. Let’s look again at the “2 in a corner” example.

2 in a virtual corner (revisited)

Pattern Example

There are two different ways that the edges around the 2 square can be coloured. They are both displayed below. (Remember: The number 2 has been coloured a faint yellow by my Slitherlinks software because it has now been “solved” and is no longer relevant to solving the rest of the puzzle. It is still there – you just need to squint! )

2 in a virtual corner

Possible Solution #1	Possible Solution #1

Most of the time, there will be some clues directly adjacent to the 2 cell that will determine which of the two possibilities is the correct one.

However, sometimes, there isn’t any clues directly adjacent. The ambiguity will need to be resolved by the trajectory of other nearby lines.

In fact, there is only one way this ambiguity could be resolved:

2 in a virtual corner (with no adjacent clues)

Only Unambiguous Solution

By the power of ‘metagaming’, you can go ahead and fill those lines in immediately!

I have previously described this kind of trick as “metagaming”, because it relies on some ‘meta-information’ that isn’t explicitly stated in the rules of Slitherlinks – that the puzzle composer implicitly promises to provide a puzzle with no ambiguity – that is only one correct solution. If this assumption doesn’t hold, then this technique is not valid.

Conclusion

Slitherlinks is a fun puzzle. If you haven’t tried it, give it a go.

I have provided a (by no means complete) list of useful patterns to look for.

I have also thrown in a couple of other techniques that can be useful: ensuring there are an even number of lines leaving any area of the grid and using meta-gaming to quickly find shortcuts to solving ambiguous cases.

I hope some of these insights can help you become a more expert Slitherlinks solver.

If you are interested how I have approached some other puzzles, please check out the puzzle links from the Start Here page.