Overview

A slitherlink puzzle consists of a grid, generally rectangular, with some numbers distributed throughout the grid. The goal is to draw a loop along the lines of grid which satisfies these rules:

1. Each number in the grid matches the number of sides of its square that the loop touches.
2. The loop is closed and does not touch itself.

Here is a sample unsolved slitherlink grid and the solution:

Fig. 1: Unsolved grid		Fig. 2: Solved grid

Slitherlink puzzles come in other shapes as well, such as triangles, hexagons, and more complicated geometric patterns, but square puzzles are by far the most common.

Strategy Summary

The key to solving slitherlink puzzles is to work several sections of the grid at once, connecting up paths as possible. As you solve, you will notice certain patterns repeat themselves, and this will allow you to get a foothold.

0s and 3s are especially useful. The loop cannot touch any side of a 0, and must touch all but one side of 3s. Hence, these represent good starting places.

Also, note that you are creating one single loop, not multiple loops. Therefore, if drawing a section would create a closed loop that doesn't include all the numbers, it can't be drawn.

Starting out

Mark off 0s

First, mark off as invalid all sides of 0.

Fig. 3: Mark up 0s

Fill in 3s

There are several important patterns. First, note that any 3 next to a 0 can be completed because there are only three remaining sides.

Fig. 4: 3 next to 0

Shared sides: If two 3s share a side, then the shared side and the opposite sides have to be part of the loop. In addition, the loop cannot extend from the shared side in a straight line:

Fig. 5: 3 next to 3

To see why this is, imagine that the shared side is not part of the loop. Then the other three sides of each 3 would be. But this would create a closed loop, encircling both 3s. Therefore, the shared side must be part of the loop. There are only two patterns where all three sides of each 3 can be drawn without a closed loop: S and Z. In each case, the shared side and the opposite sides are part of the loop, and the two extensions to the shared side can't be part of the loop.

Shared corners: If two 3s share a corner, then the opposite corners have to part of the loop:

Fig. 6: 3s share corners

To see why this is, imagine that both of the upper left 3's sides leading to the shared corner were part of the loop. This would create a corner, blocking off two of the lower left 3's sides, which isn't possible. Therefore, at most one of those two segments are part of the loop. Therefore, one of those segments is, and both of the other two segments are.

2 next to 3-3: If there is a 2 next to 3's with shared sides (so the 2-3-3 is in an L formation), then the side of the 2 opposite the 3-3 pair is part of the loop:

Fig. 7: 2-3-3 L

One side has already been ruled out. Of the other two segments touching the 2 and the upper 3, only one can be part of the loop. Therefore, the fourth side must be part of the loop.

1-3 on a border: If there is a 1 and a 3 next to each other on a border, then the 1's loop piece must be either on the border or the shared side. Additionally, the 3's border side is part of the loop:

Fig. 8: 1-3 on the border

To see why this is, imagine the shared side is not part of the loop. The other three sides of the 3 must be, and so the loop would have to extend along the border past the 1. If the shared side is part of the loop, then the loop could not extend along the border past the 1, and therefore must extend along the border past the 3.

1 in the corner

If there is a 1 in a corner, then neither of the two corner segments can be part of the loop.

Fig. 9: 1 in the corner

Remove dead-ends

There are two kinds of dead-ends. One kind is a section that only has one entry point; two of these lead to the 0 in the upper right corner.

Fig. 10: Dead-ends

The other kind is a section that leads to an intersection that already has two parts of the loop leading to it:

Fig. 11: Two segments in an intersection

Clean up dead ends again:

Fig. 12

More strategies

Complete numbers as possible

At this point, there are several cases where the number of possible remaining sides matches the number shown. For instance, the 2 in the lower right has two possible sides, as does the lower 2 in the fifth column; likewise, both the 1 in the lower left and the 1 on the right edge have one possible side.

Fig. 13

Complete the loop as possible

There are also some portions of the loop that can be completed by virtue of it only having one place to go. For instance, each end of the section next to the 1 on the right edge only has one possible neighboring section. There are a total of six sections throughout the grid we can now fill in:

Fig. 14

We can apply the above strategies iteratively:

Fig. 15		Fig. 14

Fig. 17		Fig. 18

Fig. 19		Fig. 20

Now notice the 3 on the first row. Recall that if a 1 is on a corner, neither corner segment is part of the loop. When a 3 is on a corner (as it now is), both segments must be part of the loop:

Fig. 21: 3 on a corner

Stepping back to figure 20, another approach is to consider the 1 below that 3. Notice the lower arm of the loop to the left of it. That arm must extend either to the right (below the 1) or upwards (to the left of the 1). Either of these would use up the allowed number of segments for the 1, so the other two segments can't be part of the loop:

Fig. 22: 1 with limited routes

Combining these two, we have:

Fig. 23

Again, we apply the various strategies iteratively:

Fig. 24		Fig. 25

Fig. 26		Fig. 27

Avoid closing subloops

Now we have two possible ways of completing the grid, and no remaining number clues to guide us. However, one of these results in two separate loops, which is not allowed:

Fig. 28: Two loops (invalid)		Fig. 29: One loop (solution)

Therefore, the only other possibility is the solution. We now have a closed loop that satisfies the rules.

Conclusion

This article has presented basic strategies and some common patterns for solving slitherlink puzzles. This patterns are not meant to be exhaustive, but rather presented to give you some ideas and starting points. As you solve more puzzles, you will begin to see other patterns that will simplify solving.

All slitherlink puzzles will have a unique solution which can be entirely determined by logic. Have fun!

All content copyright Paul and/or Valerie Hartzer, all rights reserved, except where indicated. Contact Paul Hartzer with comments. This file was last modified on Sunday, January 10, 2010.