Saramsha

No. 004  ·  20 September 2026

Without Lifting the Pen

Trace every line of the figure exactly once, in one unbroken stroke. Drag from corner to corner — or tap them in turn.

About this piece

Seven bridges, one walk

Königsberg in Prussia sat on both banks of the Pregel, around two islands, joined by seven bridges. The town's Sunday question was whether you could walk a route that crossed every bridge exactly once. Nobody managed it, and nobody could say why not — until Leonhard Euler answered in 1736, and in answering invented the idea of a graph.

His move was to throw away the map. Distances, shapes, the width of the river: none of it matters. All that matters is which land masses are joined, and by how many bridges. What is left is dots and lines — and a question about counting.

Here is the whole argument. Every time your stroke passes through a corner it uses two lines, one to arrive and one to leave. So lines at a corner get used up in pairs, and a corner with an odd number of lines can only be a place where you start or a place where you stop. A single stroke has one start and one stop. Therefore a figure can be drawn in one stroke only if it has no odd corners (then you finish where you began) or exactly two (you must start at one and end at the other). Königsberg has four. That settles it, forever, without trying a single route.

The odd corners in each figure here are the ringed ones — the rule is not hidden from you, it is the whole game. Notice what the counting gives you: not merely that a stroke exists, but where it has to begin.

The same counting still earns its keep. A road gritter, a postman or a street sweeper wants the shortest round that covers every street; the standard method starts by finding the odd junctions and pairing them up, which is why it is called the Chinese postman problem. And the idea has a much older cousin in south India: sikku kolam, the looped rice-flour patterns drawn at dawn around a grid of dots, are made to close in one continuous line — Euler's condition, satisfied by hand, generations before it was written down.

Everything here is drawn in your browser. The figures after the sixth are generated on the spot, each one built by taking a random one-stroke walk and then hiding it, so a solution always exists.