One Line — free draw-in-one-stroke puzzle in your browser

A shape made of dots and lines. Start on any dot and travel along the lines, covering every line exactly once, without lifting. Cover them all and the level is done; the next shape has one more line in it.

Every level is drawable by construction, not by luck: the shape is built by walking a route first and keeping the lines the walk used, so a solution existed before the puzzle did. You get sixty seconds a level, and your score is how many levels you clear. Free, no download, no sign-up.

How to play

• Tap any dot to start your line there.
• Tap a neighbouring dot to travel along the line between them. It turns yellow and is spent.
• You cannot re-use a line, but you can pass through the same dot as often as you like.
• Undo steps back one line; Start over clears the shape and gives you a fresh choice of starting dot. Neither costs anything but time.
• Cover every line before the sixty seconds run out to move to the next level.
• Your score is levels cleared, and your top 5 is kept on this device.

The trick is counting, and it was worked out in 1736

These puzzles look like they reward patience. They do not. There is a rule that tells you where to start before you draw anything, and once you know it you will clear the early levels without a single wrong turn.

Count the lines meeting each dot. A dot with an odd number of lines is the interesting kind. Here is the whole rule: a shape can be drawn in one stroke only if it has exactly two odd dots, or none at all. If it has two, you must start on one of them and you will finish on the other. If it has none, you can start anywhere, and you will finish where you started.

The reason is easier than it sounds. Every time your line visits a dot in the middle of the journey it uses two lines: one coming in, one going out. So dots in the middle always use up lines in pairs, which means an odd dot can only ever be a start or an end. You have one start and one end, so you can afford at most two odd dots, and no more.

This is Euler's rule, written down by Leonhard Euler in 1736 when he was asked whether the seven bridges of Königsberg could all be walked exactly once in a single trip. Four landmasses, and every one of them had an odd number of bridges. Four odd dots is two too many, so the walk was impossible, and the town's puzzle had an answer nobody had wanted. It is generally counted as the paper that started graph theory.

Two practical habits follow. Find the odd dots first — if there are two, one of them is your start, and picking the wrong one wastes the whole attempt. And once you are moving, do not cut off a limb: if taking a line would strand a section of the shape with no way back to it, that section is lost. The generator here always leaves a route through, so a dead end is your doing, not the puzzle's.

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• 2048 — The legendary merge puzzle. Slide the tiles, join the numbers, chase the magic 2048 tile.
• Minesweeper — The desk-drawer classic: 9×9 grid, 10 hidden mines. Read the numbers, flag the bombs, clear the board — one wrong tap and boom.
• Sudoku — Classic 9×9 logic puzzle with pencil notes and error checking. Three difficulties, every puzzle guaranteed one unique solution.

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