Slitherlink Patterns & Techniques

The 0 pattern

A cell with a 0 contributes nothing to the loop. All four of its edges are excluded. This is the first thing you should do on any puzzle: find every 0 and mark all its edges with ×.

The cascading effects are significant. Excluding the edges of a 0 means the vertices at its four corners lose possible line segments. If a vertex now has only two possible edges and needs to be on the loop (because an adjacent cell requires it), those two edges are both forced. One 0 can resolve two or three neighbouring cells before you reason about anything else.

Corner and boundary patterns

Cells on the grid boundary already have some edges “missing” in a sense — the loop cannot extend outside the grid. This constrains boundary cells more than interior cells.

3 in a corner

A 3 in a grid corner has two boundary edges. Both must be lines (if they were excluded, you could only get at most 2 lines from the two interior edges, which is less than 3). The two boundary edges are immediately forced. The corner vertex where they meet already has its 2 lines, so any other edge at that vertex is excluded.

2 in a corner

A 2 in a grid corner needs exactly two of its three available edges (the fourth is off the grid). You cannot determine which two from the 2 alone, but vertex constraints at the corner usually help. If the corner vertex has only two possible edges (the two boundary segments), either both are lines (making the 2 satisfied via the boundary) or neither is — there is no way to have exactly 1 line at a boundary corner vertex. This forces a decision quickly.

1 in a corner

A 1 in a grid corner needs exactly one of three available edges. The two boundary edges at the corner vertex must either both be lines (giving the vertex 2 lines) or both be excluded (giving it 0). If both are lines, that accounts for the corner cell's 1 plus one extra — too many. So both boundary edges are excluded, and the loop edge is one of the interior edges. This pattern immediately gives you two exclusions.

3 on a boundary (non-corner)

A 3 on an edge of the grid has three available edges (the fourth is off the grid). All three must be lines — you cannot get 3 lines from only 2 edges, so the choice is forced entirely. Mark all three as lines and exclude nothing.

The 3-3 adjacent pattern

When two 3s share an edge, that shared edge is always a line. Here is why: if the shared edge were excluded, each 3 would need all three of its other edges to be lines. But the two vertices on the shared edge would each have lines from both cells arriving from the same direction without a connecting segment, violating the vertex constraint. The shared edge must be a line.

Beyond the shared edge, the outer edges perpendicular to the shared side are also forced. If the two 3s are side by side horizontally, the left edge of the left 3 and the right edge of the right 3 are both lines. The top and bottom edges of each cell that are not shared also participate — the exact configuration depends on position, but you typically get five forced edges from this single pattern.

Diagonal 3-3 patterns (3s diagonally adjacent but not sharing an edge) also generate constraints, though they require vertex-level reasoning rather than direct edge forcing.

The 3-0 adjacent pattern

A 3 next to a 0 is one of the most productive patterns. The 0 excludes all its edges, including the shared edge with the 3. Since that shared edge is excluded, the 3 must get all three of its lines from its remaining three edges. You get three lines and four exclusions from two cells.

This pattern appears frequently on easy and medium grids. On harder grids, the 0s are rarer, but when a 3-0 does show up it is still just as powerful. Train yourself to scan for it before doing any deeper analysis.

The 1-in-corner pattern

A 1 at a grid corner forces both boundary edges to be excluded (as explained in the corner patterns section above). The interior edges of the corner cell then have exactly one line between them. But the interesting part is what happens next: the cells adjacent to the corner cell along the boundary now have one of their edges forced to be excluded (the boundary segment shared with the corner cell was excluded). This propagates along the boundary.

If a 1-in-corner is adjacent to another low clue (0 or 1) along the boundary, the cascade of exclusions can extend several cells deep. Look for these chains on medium and harder puzzles.

Edge counting

This is not a flashy pattern, but it is the technique you will use most. For every numbered cell, maintain a running count: how many edges are already lines, how many are excluded, how many are undetermined. When the lines equal the clue number, all undetermined edges are excluded. When the exclusions equal 4 minus the clue, all undetermined edges are lines.

Edge counting sounds mechanical, and it is. But on a 10×10 grid with 45 or 50 clue cells, you can resolve a dozen edges per pass just by scanning each cell and updating counts. It is the engine that converts the fancy patterns above into actual progress.

Our interface helps: clue numbers turn green when satisfied and red when over-constrained. If a number stays grey, the cell still has undetermined edges. If it turns copper, some edges are placed but the count is not yet met.

Loop closure reasoning

On easy and medium grids, the number clues alone usually determine the entire loop. On hard grids and above, you hit points where no single clue forces an edge. That is where the single-loop constraint becomes your primary tool.

The rule is: the lines must form exactly one closed loop. If you have a partial loop (a connected path with two open endpoints), connecting those endpoints would close the loop. If doing so would leave other placed lines disconnected, that connection is illegal and the edge must be excluded.

The reverse applies too. If excluding an edge would make it impossible for the existing lines to ever form a single connected loop (because it would isolate a section), that edge must be a line.

Tracking this mentally requires knowing which vertices are connected by the partial loop. On 10×10+ grids, it helps to trace the path periodically and identify the two open endpoints. Much of the endgame on expert and Einstein puzzles comes down to: does this edge connect or disconnect the path?

Region and parity analysis

This technique rarely appears below expert difficulty, but when it applies, it resolves edges that nothing else can.

Any closed loop on a grid divides the cells into two groups: those inside the loop and those outside. If two adjacent cells are on the same side (both inside or both outside), their shared edge is not part of the loop. If they are on opposite sides, their shared edge is part of the loop. You can sometimes determine the inside/outside status of cells by counting known loop crossings from the cell to the grid boundary.

Parity arguments extend this. If a row of cells between two known loop segments must have an even or odd number of line edges to maintain consistent inside/outside status, you can force or exclude edges purely from parity constraints. This shows up on Einstein grids where clues are sparse and direct deduction stalls.

Common mistakes

Focusing only on clue cells. Blank cells are just as constrained by vertex and loop rules. A vertex between four blank cells still needs 0 or 2 lines. Re-check vertices after every placement, not just cells with numbers.

Not using exclusions early enough. New players draw lines but forget to mark × on excluded edges. Without exclusions, edge counting is far less effective because you do not know how many undetermined edges remain.

Trying to guess rather than deduce. If you find yourself guessing which edge is a line, stop and look for a different cell where the logic is clearer. On a well-made puzzle, there is always at least one determinable edge. Guessing often leads to contradictions several moves later, and backtracking in slitherlink is painful because edge states interact in chains.

Recommended practice path

Start with easy 5×5 puzzles to learn the 0 pattern, corner 3s, and basic edge counting. These should take 2–3 minutes once the patterns click.

Move to medium 7×7 when easy feels automatic. Medium introduces the 3-3 and 3-0 patterns more regularly and requires systematic edge counting across the whole grid.

Hard 10×10 is where loop closure reasoning becomes necessary. Clues alone will not finish the puzzle. You need to track the partial loop and reason about connectivity.

Expert 12×12 and Einstein 14×14 demand all the above plus region parity and long-range chain reasoning. Take your time with these. An hour on an Einstein puzzle is normal.