Slitherlink
Connect dots to draw a single unbroken loop guided by surrounding cell numbers.
Basic Game Premise & Rules
MechanicsSlitherlink is an untimed, deterministic pencil-and-paper logic puzzle played on a grid. Every valid puzzle has exactly one unique solution that can be discovered through rigorous deductive reasoning alone, without trial and error or guessing.
- Draw a single continuous, non-intersecting closed loop connecting horizontally and vertically adjacent dots.
- A number inside a cell indicates exactly how many of its four boundary edges are part of the loop.
- Cells without numbers may have any number of edges (0, 1, 2, or 3) on the loop.
- The loop cannot branch, cross itself, or form multiple separate loops.
- Every board has exactly one unique solution, deducible through pure spatial deduction without guessing.
History & Origins
Origins & EvolutionSlitherlink was invented in Japan by Nikoli and debuted in *Puzzle Communication Nikoli* #26 in June 1989. It rapidly rose to become Nikoli's quintessential loop puzzle, earning praise from mathematicians and puzzle lovers worldwide for its pristine elegance.
Throughout the 1990s and 2000s, it was syndicated globally in major newspapers—including *The Times* (London) and *The Nikkei* (Tokyo)—under titles such as **Fences** and **Loop the Loop**. Unlike grid puzzles that fill cells with numbers, Slitherlink operates along the discrete edges of the grid graph, creating an intuitive relationship between local clue constraints and the global topology of a closed Jordan curve.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Slitherlink formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Jordan Curve Theorem & Cut Parity
Any valid Slitherlink loop partitions the plane into an 'inside' and an 'outside' (Jordan Curve Theorem). Consequently, any continuous path from the exterior to an interior cell must cross the loop an odd number of times. For any cut (partition of vertices into S and V \ S), the number of loop edges crossing the cut must be even, which provides a powerful topological parity invariant.
Computational Complexity: NP-Completeness
In 2000, Takayuki Yato proved that deciding the solvability of a general Slitherlink puzzle is NP-complete. The proof constructs a reduction from Planar 3-SAT, using loop segments as Boolean wires, clause checkers, and turn gadgets. Even when restricted to grids containing only 0s, 1s, 2s, and 3s, the problem remains NP-complete.
Eulerian Degrees & Vertex Conservation
At every grid dot (vertex v), the degree of incident loop edges must be either 0 (unvisited) or exactly 2 (pass-through). Degrees of 1 (dead ends) and 3 or 4 (branching or self-intersection) are strictly forbidden. This 2-factor constraint enables fast Boolean Satisfiability (SAT) encodings using XOR/cardinality networks.
References & Further Reading
Bibliography- Yato, T. (2000). 'Complexity and Completeness of Finding Another Solution and Its Application to Puzzles'. Information Processing Society of Japan (IPSJ).
- Nikoli (1989). Puzzle Communication Nikoli #26. Tokyo: Nikoli Publishing.
- Demaine, E. D., & Hearn, R. A. (2009). Games, Puzzles, and Computation. CRC Press. ISBN: 978-1568813226.
- Friedman, E. (2002). 'Pencil Puzzles: A Survey of NP-Completeness'. Stetson University Mathematics.
Free Printable PDF Booklet
Download our beautifully formatted, high-resolution A4 puzzle booklet. Generated directly from the handcrafted levels on AZTest, it contains 50 puzzles ordered from gentle introductory boards up to master-level challenges, followed by complete solutions at the back.
| Difficulty Stage | Grid Size | Description | Puzzles |
|---|---|---|---|
| Intro | 5×5 | Gentle loops introducing corner 0s and adjacent 3-3 pairs. | 3 |
| Easy | 5×5 to 6×6 | Standard deductions: corner 3s, diagonal 3-3s, and end-trapping. | 12 |
| Medium | 7×7 to 8×8 | Jordan inside/outside parity reasoning across larger boards. | 12 |
| Hard | 8×8 to 10×10 | Cut-set analysis and long loop avoidances. | 12 |
| Expert | 10×10 | Master grids requiring global loop connectivity synthesis. | 11 |
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SeriesWolves & Sheep
Draw a Slitherlink loop that fences the gentle sheep in and keeps the fierce wolves out.
Masyu
Draw one closed loop through cell centres that rigorously obeys the black and white pearls.
Nurikabe
Shade the sea to leave numbered white islands of exact sizes in one connected ocean.
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Shade runs of cells according to row and column number clues to reveal a hidden picture.
Themed Nonogram
Solve color picture cross puzzles that blossom into vibrant pixel art upon completion.
Hitori
Shade the repeated numbers until every row and column has unique digits, keeping all white cells connected.
Kurodoko
Shade cells so every numbered circle sees exactly that many white cells in its row and column.
Mosaic
Shade cells guided by numbers that count how many cells around them are filled, revealing a hidden picture.
Binairo
Fill every cell with blue or white circles: never three in a row, every line half and half.
Heyawake
Shade rooms according to their clues without isolating unshaded cells or leaving long straight corridors.
Searchlights
Place light beams in white cells so that every numbered black post sees its exact clue count.
Light Up
Place light bulbs to illuminate every white cell without any two shining on each other.
Calcudoku
A Latin square where every outlined cage satisfies its arithmetic target and operator.
Skyscrapers
Place towers of heights 1 to N so that each outside clue sees the exact count of buildings.
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Futoshiki
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Divide the grid into polyomino blocks where each cell's number equals the area of its block.
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Partition the grid into rotational 180° symmetric galaxies, each centered on a single dot.
Shikaku
Cut the whole grid into rectangles and squares, each containing exactly one number: its area.
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Draw borders to partition all white cells into perfect four-cell L-shapes.
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Connect pairs of matching numbers with continuous non-crossing paths covering the grid.
Bridges
Connect the island network with straight horizontal and vertical bridges into one united world.
Slant
Draw a diagonal slash in every cell matching the corner numbers, without ever closing a loop.
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Pitch a tent beside every tree with no two tents touching, matching row and column counts.
Marupeke
Place circles and crosses in every cell without creating three in a row in any direction.