One Stroke

Touch a starting node, then drag along edges to neighbouring nodes. Every edge must be traversed once. Cannot retrace edges. Cannot lift the finger mid-stroke. The level clears when all edges are drawn.

One Stroke is a graph-traversal puzzle where you have to draw every line in a figure without lifting your finger and without retracing any line. The math behind it is Euler's bridges-of-Königsberg problem from 1736, and the puzzles are essentially small-scale Eulerian-path problems.

What works is the constraint clarity. The rules are simple: every edge must be drawn exactly once; you cannot lift your finger mid-stroke. The path can revisit nodes but cannot revisit edges. These rules are easy to internalise and the format depends on figuring out which nodes to start from. There is a mathematical theorem (an Eulerian path exists if and only if the graph has zero or exactly two odd-degree nodes) that lets you instantly see whether a level is solvable and where to start, and the game does not tell you this until you have figured it out through play.

One hundred levels across four difficulty tiers. The early levels are small graphs with obvious paths. The mid tiers introduce graphs where the start node matters significantly, and the late tiers add coloured edges that must be drawn in specific sequences. The progression earns its difficulty.

Tested over a week of commute sessions. Touch with finger-trace is the optimal input. Mouse-drag works fine. Stylus would probably be the best input but I do not have one to test. Keyboard input is not supported.

Where the design impresses me is the way it teaches Euler's theorem implicitly. By tier two, players are already developing the intuition that nodes with odd degree must be start or end points. By tier three, players are using this intuition without naming it. This is excellent pedagogy through gameplay, and it earns the four-and-a-half stars on the strength of teaching-by-doing alone.

Where I would push back is the absence of a graph-editor mode for creating custom puzzles. The format would thrive on player-created levels with community ratings. The fixed-puzzle catalogue is large enough to last weeks but will eventually be exhausted.

Four-and-a-half stars. Strong graph-traversal puzzle that teaches mathematical thinking without ever making it feel academic. Recommended for puzzle players who enjoy elegant constraint systems.