How to solve one-line drawing puzzles (and spot the impossible ones)

By The Boring Crew · Updated 2026-11-06

A one-line drawing puzzle asks you to trace a whole figure without lifting your pen and without going over any line twice. One simple counting rule tells you, before you start, whether it can be done and where to begin.

The rules

You get a figure made of dots and the lines between them. Draw every line exactly once, in one go. You can pass through a dot as often as you like, but you can’t lift your pen, jump to another dot, or trace a line a second time. Some figures simply can’t be drawn this way, however clever you are.

The odd-dot rule

At every dot, count how many lines meet there. Call a dot odd if that number is odd (1, 3, 5…) and even if it is even (2, 4, 6…). Then count the odd dots:

Odd dots	Can it be drawn in one line?	Where to start
0	Yes	Anywhere. You will finish where you started.
2	Yes	At one odd dot. You will finish at the other one.
4 or more	No, never	It can’t be done. Each extra pair of odd dots needs one more pen lift.

You never get exactly 1 or 3 odd dots: every line has two ends, so the total of all the counts is always even, and the odd dots always come in pairs.

Two examples

The envelope house: can be done

The roof tip has 2 lines, the top corners have 4 each, and the two bottom corners have 3 each. That is exactly two odd dots (orange), so start at one bottom corner and you will end at the other. The crossing in the middle is not a dot, so it doesn’t count.

A square with both diagonals: can’t be done

Take the roof away and every corner has 3 lines. Four odd dots means it is impossible in one line. You would need at least two strokes.

Once you know it can be done

Start at an odd dot if there are two. Starting anywhere else is the most common way to get stuck.
Don’t cut yourself off. Before you use a line, check that it isn’t the last link to a part of the figure you haven’t drawn yet. Leave such “bridge” lines for later if you have another choice.
Finish small loops as you pass them. A loop you skip now often becomes a loop you can’t reach later.
If you get stuck, start again from the other odd dot. The route often looks very different from that end.

Where the rule comes from: in 1736 Leonhard Euler asked whether you could walk over each of the seven bridges of Königsberg exactly once. Counting the bridges at each piece of land showed it was impossible, and that idea started graph theory. A one-line route is still called an Euler path.

Using it in class

This puzzle works well with students from about age 8. Let them try a few figures first, then ask them to count lines at each dot and find the pattern themselves before you show the table. Every player gets the same five figures each day in Unbroken, so a whole class can play the same set on their phones or on a projector. It is free, runs in the browser, and needs no sign-up.

Play Unbroken, the free one-line drawing puzzleFive new figures a day and 600 levels for ages 6 to grown-up. Some can’t be done. Spot them.

Like puzzles you trace with a finger? Try Knots, where you untangle crossing lines, or Slice, where you cut a shape exactly in half.