One Line Puzzles
Trace every line of the figure without lifting your finger, and never go over the same line twice. Euler settled this one in 1736.
Hints available. Play as many boards as you like.
Ready when you are.
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Scan for the same board — no hints.
How to play
Drag from node to node along the printed lines. Each line darkens as you trace it, and a line already traced cannot be used again — the board refuses the drag rather than letting you build a stroke that cannot finish.
You may pass through the same node as often as you like. It is the lines that are used up, not the dots, and most figures need you to cross a busy junction three or four times.
A figure with two ringed nodes is an open one: begin at a ring and you will finish at the other. A figure with no rings is a closed loop and any node will do, because wherever you start is where you will end up.
Backspace, Delete or Z undoes the last line. Arrow keys walk through the legal next lines and Enter traces the highlighted one, so the whole puzzle is playable without a pointer.
Strategy
Count the lines at each node before you start. Nodes with an odd number of lines are the only places an open figure can begin or end, and there are never more than two of them — that is Euler's rule and it is the entire theory of this game.
On a closed figure, every node has an even count, so any node you leave you can always leave again. That is why starting position never costs you a closed board and frequently costs you an open one.
Do not cut off a limb. If tracing a line would leave a group of untraced lines with no route back to where you are standing, that group is stranded — the board will still let you finish the near side and then strand you.
Save the spur for last. A dead-end line hanging off the figure has to be traced on the way to the finish, never in the middle, unless the node it hangs from is where you intend to stop.
When you are stuck, undo to the last junction where you had a real choice rather than starting over. The mistake is almost always one turn, and the fifteen lines before it were fine.
Difficulty and variants
Easy figures hold 12 lines and are always closed loops, so any starting node works. Medium holds 20 and hard 30, both open figures with exactly two legal starting points among their twenty-odd nodes. On a wide screen the same three levels use a larger lattice and 16, 26 and 36 lines.
Questions, answered
Can every figure really be drawn in one stroke?
Every figure here can. Euler proved that a connected figure is drawable in one unbroken stroke exactly when it has either no odd nodes or precisely two, and the generator repairs each board until it satisfies that before offering it to you.
Why does it matter where I start?
On an open figure the two odd nodes must be the start and the end. Beginning anywhere else leaves an odd node in the middle of your route with one line too many, and the stroke is doomed however cleverly you continue.
Can I go through a node more than once?
Yes, as many times as its lines allow. Only the lines are consumed. A node with six lines will be visited three times in every solution.
Is there more than one answer?
Usually many. Unlike the deduction puzzles here, this one is not asking you to recover a hidden arrangement — any route that uses every line exactly once counts, and the board checks that property rather than comparing you to a stored solution.
What is the Königsberg bridge problem?
The question that started all of this: could you walk the seven bridges of Königsberg, each exactly once? Euler showed you could not, because all four landmasses had an odd number of bridges. Two odd nodes are permitted, four never are.