One Line Draw

What is One Line Draw?

A shape made of dots and connecting lines appears. Trace every line exactly once without lifting your finger and without going over any line twice. You can pass through a dot as often as you like. When every line is covered, the level is done.

The puzzle is much older than the app. It is the Seven Bridges of Königsberg, the problem Leonhard Euler solved in 1736 and, in doing so, invented graph theory. The people of Königsberg wanted to know whether they could walk a route crossing each of the town's seven bridges exactly once. Euler proved they could not, and the reason he gave is the same rule that decides whether any level here is solvable.

The rule that tells you if it can be done

Count the lines meeting at each dot. A dot with an odd number of lines is an odd point. A shape can be drawn in one stroke only if it has exactly zero or two odd points.

• Zero odd points: you can start anywhere, and you will finish where you started.
• Two odd points: you must start at one of them and you will finish at the other.
• Four or more odd points: impossible in a single stroke, no matter how clever you are.

That is the entire theory. Every level in this game is generated and then verified against that rule before it is given to you, so nothing here is unsolvable.

How to play

1. Tap a dot to start, or drag across the shape.
2. Move to a connected dot to draw that line. It fills in as you go.
3. Each line can only be used once. Dots can be revisited freely.
4. Cover every line to clear the level. Undo steps back one line at a time.

How to solve them faster

• Find the odd points first. If there are two, one of them is your only possible starting dot. That single observation removes most of the guesswork.
• Do not strand a section. The most common failure is using the only line into an area before you have finished inside it.
• Save bridges for last. A line that is the sole connection between two halves of the shape should be crossed once, late, and only when the first half is complete.
• Work the outside in. Perimeter lines usually have fewer alternatives than the ones in the middle.
• Undo is free. There is no move limit and no timer, so exploring a wrong branch costs nothing but a tap.

Difficulty and levels

Easy uses a three by three lattice, normal a four by three, and hard a four by four with diagonals in play. Levels grow denser as you clear them, from around seven lines up to twenty-six. They are generated fresh rather than hand-built, so they do not run out, and the parity check runs on every one before you see it.

Your furthest level is saved per difficulty in your own browser. There is no account, no timer and no way to lose, which makes this one of the better games here for playing while thinking about something else.