"Draw this house in one go: don't lift your pencil, and don't go over any line twice." It's an old folk puzzle that many grandparents remember from their own childhood, and it still delights children today. Some shapes can be drawn in one stroke, some can't, and there's a neat secret that tells you which is which.
One-stroke puzzles are excellent for spatial thinking and planning. Children have to picture a route through a shape before they commit to it, and when they get stuck, they learn to think about where they started.
The rules
- Keep the pencil on the paper from start to finish.
- Draw every line of the shape.
- Never go over the same line twice. Crossing a line at a point is fine; travelling along it again is not.
Give your child a few shapes to try: a square, a triangle, a square with one diagonal, then the classic house with a roof and two diagonals across the walls.
Where you start matters
Most children start at the top of the house, and get stuck. Then they try again from another corner, and get stuck again. Then, often by accident, they start at one of the bottom corners and it works.
That's the big discovery: for some shapes, only certain starting points work. Ask your child to mark a dot where they started each time, and to colour the dot green when it worked. After a few tries, a pattern appears.
The counting trick
For children around 8 and older, here's a lovely secret to discover together. Look at each corner or crossing point in the shape and count how many lines meet there.
| What you count | What it means |
|---|---|
| Every point has an even number of lines (2, 4, 6...) | You can start anywhere, and you'll finish where you started |
| Exactly two points have an odd number of lines (1, 3, 5...) | You must start at one of those two points and finish at the other |
| More than two points have an odd number | The shape can't be drawn in one stroke |
Try it on the house: the two bottom corners each have three lines (odd), and every other point has an even number. So you must start at one bottom corner and finish at the other, exactly as your child discovered.
Don't hand over the table at the start. Instead, ask: "Let's count the lines at each corner. What's different about the corners where it worked?" The discovery is the best part.
Shapes to try
Draw these on paper, or make them with lolly sticks or straws:
- Easy: a triangle, a square, a star made of five lines, a figure of eight.
- Medium: a square with one diagonal, the house, an envelope with the flap closed.
- Tricky: a square with both diagonals but no roof (can it be done?), two squares side by side sharing an edge.
- Impossible: some shapes really can't be done. Finding out why is just as satisfying as solving one.
Use a different colour pencil for each attempt, or trace over a printed shape with a finger before drawing, so mistakes don't build up on the page.
Let children invent shapes
Once your child understands the game, invite them to draw shapes for you:
- "Can you draw a shape that I can do in one stroke?"
- "Can you draw one that's impossible? How do you know?"
- "Can you change one line to make the impossible shape possible?"
Inventing and checking shapes turns a puzzle into real reasoning. Older children can use the counting trick to check their shapes before testing them.
In Dologi, "One stroke" puzzles show a few line figures and ask children to find the one that can be drawn in one go, and "Trace in one go" puzzles ask them to trace a whole figure without lifting their finger.
In short
- One-stroke puzzles: draw every line without lifting the pencil or going over a line twice.
- Where you start matters; mark your starting dot each time and look for a pattern.
- Count the lines at each point: all even, start anywhere; two odd, start at one of them; more than two odd, impossible.
- Let children invent shapes and check them, possible or impossible.
Read more: What is spatial thinking? Building blocks, mazes and paper folding for children