Endless mode
A curated progression of grids that introduces mechanics one at a time: walls, checkpoint caps, ice, veils, switches, portals. Past level 60 the live generator takes over.
A hamiltonian path game in your pocket.
monostroke is a single line drawing puzzle built on the hamiltonian path. Trace one stroke that covers every cell of a small grid to earn three stars. Free on iOS and Android.
Background
What is a hamiltonian path?
A hamiltonian path is a path through a graph that visits every vertex exactly once. Named after the 19th-century Irish mathematician Sir William Rowan Hamilton, the concept appears across recreational math, computer science, and a small but loyal corner of puzzle design.
Translated to a grid-based puzzle: a hamiltonian path is one continuous line that touches every cell exactly once, never crossing itself. If the line ends where it began — a closed loop — it's called a hamiltonian cycle. Hamiltonian paths are deceptively hard to find; deciding whether a graph has one is, in general, computationally difficult. On a small puzzle grid, it becomes a tractable, satisfying problem.
In monostroke
Three stars require a hamiltonian path.
Every monostroke level is a small grid (typically 3×3 to 6×6) with one or two endpoints. Press an endpoint and drag the line one cell at a time — up, down, left, or right. The line cannot cross itself. The goal is simple: make the line as long as possible.
Star grading is graded against the puzzle's longest valid path. Three stars are reserved for the maximum — a true hamiltonian path that covers every cell. Two stars for a substantially long path, one star for a valid but short one. Every monostroke puzzle is verified by the generator to have at least one hamiltonian path before shipping to the player.
Challenge worlds
Three thirty-level worlds built around variations: Blueprint (closed-loop endpoints — hamiltonian cycles), Mirror (reflected grids with synchronised halves), Tally (numbered checkpoints visited in sequence).
Gauntlet
Five floors, each demanding a 3-star hamiltonian path on a fresh procedurally-generated puzzle. One miss ends the run; you keep the shards banked from earlier floors.
Adjacent puzzles
Where hamiltonian paths show up.
Hamiltonian paths and cycles are at the centre of a small but real corner of puzzle design and recreational math. monostroke is one mobile-native expression of that family.
- Hamiltonian cycle puzzles — closed-loop variants where the line returns to its start. monostroke's Blueprint world is built entirely on these.
- Knight's tour — a classical chess puzzle: visit every square of a chessboard with a knight, exactly once. A constrained hamiltonian path on the knight-move graph.
- Eulerian paths — the cousin: visit every edge exactly once. The Königsberg Bridges problem is the foundational example. Different graph property, similar feel.
- Travelling salesman — hamiltonian path with a cost function: visit every city, but minimise total distance. monostroke flips this — maximise length, not minimise.
- Snake-style trail puzzles — many "snake puzzle" mobile apps are also implicitly hamiltonian-path puzzles, even when they don't use the term.
FAQ
Common questions.
What is a hamiltonian path?
A hamiltonian path is a path through a graph that visits every vertex exactly once. In a grid-based puzzle, that means drawing one continuous line that touches every cell exactly once without crossing itself. A hamiltonian cycle is the same idea but the start and end cells are the same — a closed loop.
Is monostroke a hamiltonian path game?
Yes. monostroke is a single line drawing puzzle where the maximum-score solution is a hamiltonian path on the playable cells of each level. Three stars require visiting every required cell with one unbroken stroke. Lower-scoring partial paths are still valid and earn one or two stars.
Do all monostroke puzzles have a hamiltonian path?
Yes. Every monostroke puzzle is generated or hand-authored to guarantee at least one hamiltonian path exists. The procedural generator validates this property before shipping a puzzle to the player.
How is hamiltonian path different from eulerian path?
An eulerian path visits every edge of a graph exactly once; a hamiltonian path visits every vertex exactly once. In a grid game, eulerian-style puzzles have you cross every wall or boundary; hamiltonian-style puzzles have you fill every cell. monostroke is firmly in the hamiltonian camp — cells, not edges.
Where can I play hamiltonian path puzzles on mobile?
monostroke is one of the few mobile games built specifically around hamiltonian paths. Adjacent options include certain hexagonal puzzle apps, some Logic Square apps, and academic puzzle anthologies. monostroke is purpose-built around the mechanic with progressive complexity (walls, ice, switches, portals, mirrors) layered on top.
Where can I download monostroke?
monostroke is available on the App Store (iPhone, iPad) and Google Play (Android). Both versions are free with cosmetic-only in-app purchases.
Get monostroke
Free on iOS and Android.
No account. Plays offline. The full puzzle set is free.