Numberlink
Connect pairs of matching numbers with continuous non-crossing paths covering the grid.
Basic Game Premise & Rules
MechanicsNumberlink 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.
- Connect each pair of matching numbers with a single continuous line of horizontal and vertical steps.
- Lines cannot cross each other, branch, or share any cells.
- Lines must start and end at their matching numbers, and cannot pass through other numbers.
- Full Grid Coverage: Every cell on the board must be traversed by exactly one line.
- Every puzzle has a unique solution discoverable through topological path reasoning.
Rules by Example
See how each rule looks when done correctly versus when violated:
Connecting Matching Numbers
Connect each pair of matching numbers with a single continuous line of horizontal and vertical steps. Lines must never cross each other.
Lines and Endpoints
Numbers are endpoints only. Lines cannot branch, fork into T-junctions, or pass through another number.
History & Origins
Origins & EvolutionNumberlink was invented by Nikoli and debuted in *Puzzle Communication Nikoli* #27 in September 1989. It originated from printed circuit board (PCB) design problems, where electrical engineers needed to route wires between terminal pins on a single-layer board without wire crossings.
In 2012, Big Duck Games released *Flow Free* on mobile platforms, turning Numberlink into one of the most downloaded mobile games in history. In Nikoli's rigorous formulation, all cells on the board must be filled, ensuring a mathematically airtight, unique solution without ambiguous slack.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Numberlink formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Edge-Disjoint & Vertex-Disjoint Paths Problem
Numberlink directly models the 2D Disjoint Paths Problem on planar grid graphs: given k pairs of terminals (s_i, t_i), find k mutually vertex-disjoint simple paths connecting each pair. In general graphs, this is one of Karp's 21 NP-complete problems.
VLSI Circuit Routing & Topological Obstacles
In electronic design automation (EDA), Lee's routing algorithm and A* maze routers solve single-path connections. In Numberlink, routing path 1 acts as a dynamic obstacle for path 2, creating topological homotopy constraints.
NP-Completeness via Planar Graphs
In 2010, I. Kotsireas and D. Karapetyan proved that Numberlink is NP-complete. The full-grid coverage rule forces dead ends to extend, meaning empty spaces cannot be bypassed.
References & Further Reading
Bibliography- Kotsireas, I. S., & Karapetyan, D. (2010). 'On the NP-completeness of the Numberlink Puzzle'. International Journal of Computer Mathematics.
- Nikoli (1989). Puzzle Communication Nikoli #27. Tokyo: Nikoli Publishing.
- Robertson, N., & Seymour, P. D. (1995). 'Graph Minors. XIII. The disjoint paths problem'. Journal of Combinatorial Theory, Series B.
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 | Simple 3-pair connections demonstrating non-crossing paths. | 3 |
| Easy | 5×5 to 6×6 | Corner pairs and edge hugging routes. | 12 |
| Medium | 7×7 to 8×8 | Five and six-pair networks requiring full-grid cell coverage. | 12 |
| Hard | 8×8 to 10×10 | Deep serpentine paths that weave around terminal points. | 12 |
| Expert | 10×10 | Master routing challenges with dense numbers and zero wasted space. | 11 |
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