Slitherlink techniques: patterns and strategy

Technique 1 · basic

Zeros: an x on all four sides

A 0 is the easiest number on the board and the best place to start. The loop uses none of its four sides, so all four get an x right away. That sounds too simple to count as a technique, but a 0 does more than clear one tile: each of its x marks is also missing from the tile next door and from the two dots at its ends. A 0 next to a 3, a 0 on the rim, or two 0s close together usually settle several more edges within a move or two.

Before anything else, scan the whole board for 0s and cross out their sides. Then look at what each x did to its neighbors. That one sweep often decides more edges than any later step. A 0 on the rim also rules out the next rim edge on each side: the rim dot there is left with a single open edge, and a line can't end at a dot.

Technique 2 · basic

The dot rule: two lines or none

Every dot on the board ends up with exactly two lines or with none, because the loop passes through a dot or stays away from it, and it never stops, branches or crosses itself. That gives three quick moves. A dot with two lines already gets an x on its other edges. A dot with one line and only one open edge left must continue along that edge. A dot with no line and only one open edge can't take a line there at all, since the line would have nowhere to go next, so that edge gets an x.

The dot rule is what carries a line forward after every other technique. Dots on the rim have three edges and the four board corners have only two, so lines there run out of options fast. Whenever you draw a line, look at both of its ends.

Technique 3 · basic

Corner 3 and corner 1

A board corner is a dot with only two edges, so the loop either turns there, using both, or stays away. Now put a number in the corner tile. A 3 uses three of its four sides, so at least one of its two rim sides is a line, and a line at the corner can't stop: both rim sides are lines. A 1 is the opposite case. If it used one rim side, the corner would force the other one too, and that would be two lines. So a 1 in a corner gets an x on both rim sides.

This works on any dot, not only the four corners. Once x marks leave a dot with only the two sides of one number open, that dot acts as a board corner for the number: a 3 there takes both sides as lines, and a 1 gets an x on both.

Technique 4 · basic

A 3 next to a 0

A 3 needs three of its four sides. When the side it shares with a 0 has an x, only three sides are left, so all three are lines. The pattern goes further than that tile. The two lines along the sides of the 3 end at the 0's corners, and each of those dots now holds one line with the 0's edges crossed out, so the dot rule pushes both lines on at once, away from the 0. In this board the 3 sits on the rim, so its third line runs along the border.

The same reasoning works for any number whose x marks leave it exactly as many open sides as it needs: a 2 with two crossed sides, or a 1 with three. Count the open sides of every number after each round of x marks.

Technique 5 · intermediate

A 2 in a corner, and 1s along the edge

A 2 in a corner tile has only two ways to work. Either the loop turns at the board corner and uses both rim sides, or it stays out of the corner and uses the two inner sides. Look at the rim edge just past the 2 on each side. In the first case the line along the rim must carry on along that edge. In the second case the inner side meets the rim at that same spot, and the rim dot has nowhere else to send it. Both cases put a line there, so draw both of those rim edges without knowing which case is true.

A cousin of this pattern is two 1s side by side along the rim. The rim dot between them has three edges, and a line through it using the edge between the two tiles would give one of the 1s two lines. So the edge between two 1s on the border gets an x.

Technique 6 · intermediate

Two 3s side by side

Two 3s that share a side are one of the strongest slitherlink patterns. Their two far sides, the ones parallel to the shared edge, are always lines. The edges that would continue the shared edge, straight beyond it at both ends, always get an x. Try it: if either far side were empty, that 3 would need both of its sides along the pair plus the shared edge, and one of the dots at the end of the shared edge would end up with three lines or a dead end.

The shared edge itself is a line too in every real puzzle: without it the two 3s would be wrapped by one small loop around just those two tiles. The solver settles it a little later with the other rules, but you can draw it now. A row of three or more 3s works the same way: every edge between them, and both ends, become lines.

Technique 7 · intermediate

Two 3s on a diagonal

Two 3s that touch only at a corner fix their outer corners. The upper 3 takes the two sides that face away from the lower one, and the lower 3 does the same on its far side, so four lines appear at once. Why: if the upper 3 left out one of its outer sides, it would use both sides that meet at the dot it shares with the other 3. That dot would then be full, the lower 3 would lose two sides and be left with only two. At each outer corner the two lines meet and turn, so the other edges at that dot get an x.

The pattern also works through a chain of 2s. If a 3 and another 3 sit on the same diagonal with only 2s between them, the outer corners of both 3s are lines, just as if they touched. Look for diagonals of 3s and 2s across the board before you start tracing.

Technique 8 · intermediate

No small loops

There is only one loop, so the line can't close into a ring until the very last edge of the puzzle. Whenever an open edge would join the two ends of the same line, ask whether that would finish the whole board: every number exact and no other lines anywhere. If not, the edge gets an x. This catches U shapes around a single tile, and longer lines whose two ends have drifted close to each other.

Follow each line from end to end now and then: two ends that belong to the same line are easy to miss when they are far apart along it but next to each other on the board. Combine this with the dot rule: the x often leaves one of the ends with a single way on.

Technique 9 · advanced

Parity: inside and outside

Any closed loop that doesn't cross itself splits the board into inside and outside. That is the Jordan curve theorem, and it gives slitherlink a counting rule. Walk down one column of tiles from above the board to below it: you start outside and end outside, so you cross the loop an even number of times. Those crossings are the column's horizontal edges, top rim to bottom rim. So every column, and every row, holds an even number of loop edges.

When all but one edge of a row or column is settled, count its lines: an odd count means the last edge is a line, an even count means an x. The same idea works tile by tile. The area beyond the rim is outside; a tile across an x from an outside tile is outside too, and a tile across a line from it is inside. Two tiles that are both inside or both outside never have a line between them.

Common mistakes

• Drawing lines and skipping the x marks. Most techniques start from an x: a crossed edge is what turns a dot into a corner or leaves a 3 with exactly three sides. Mark every edge you rule out.
• Reading a 3 next to a 0 only on its own tile. The lines it gets point somewhere. Follow both ends with the dot rule before you look elsewhere.
• Drawing the middle of two side-by-side 3s and stopping. The far sides and the x marks beyond the shared edge are the part that spreads. Draw all of it.
• Closing a loop too early. Before you draw an edge that joins two line ends, check that they are not the two ends of the same line, unless that edge finishes the board.
• Guessing when you're stuck. Count the edges in each row and column first. Parity often settles the one edge everything else is waiting for.