Draw it without lifting

What you need

• Paper
• A pencil

The rule

Never lift the pencil. Never go over a line twice.

How to do it

1. Mark four corners of a square with a roof point above it.
2. Sketch the intended square, two roof sides and both square diagonals lightly as a guide.
3. Count the lines meeting at each corner without tracing the route yet.
4. Start a darker route at either lower square corner.
5. Follow unused lines continuously until every guide line has been covered once.
6. Check that the route ends at the other lower corner.

Why it works

Imagine each corner as a meeting place and each line as a path. At an ordinary middle stop, your route needs one line to enter and another to leave. That uses paths in pairs. A corner with an odd number of paths therefore needs to be the start or finish in this connected drawing. The house has two such lower corners. Planning from one of them makes a complete one-pass route possible without retracing an edge.

Make it harder

• Try starting at an upper corner and explain the problem.
• Find a second valid route from the same lower corner.
• Design another connected picture with exactly two odd-path corners.

Ask Taz

• Why do middle visits use lines in pairs?
• Why must an odd-path corner be an endpoint?
• Why does adding one line change whether a route works?

Ask Taz

For grown-ups

Help the child separate planning from the final continuous drawing. A faint guide is preparation, not a completed challenge attempt. The useful explanation is about routes and connections, rather than neatness. Ask at dinner: “Why did choosing a different starting corner change whether you could finish?”

Taz is an AI and can be wrong, so stay close by. Read how safety works on Taz.

Show us yours

Film your hands and what you made, not your face or name. Ask a grown-up before posting anywhere else.

Questions parents ask

Does the result need to look polished?

No. Begin with a small attempt and keep the first version so you can explain a change. The stated time is an estimate, and a careful observation or unfinished draft can still give you something useful to discuss.

How can a parent help without taking over?

Help with the space and listed preparation, then ask the child to describe their next choice. Let the rule guide the attempt instead of correcting every line or deciding the finished design for them.

Do we have to upload the work?

No. Showing it at home is enough. Online sharing remains optional and follows the privacy and grown-up permission line above.

More to try

See all Taz Challenges, or look through the science labs. One place for every why. Ask for early access at aitaz.com

Try it in 3D: Graph theory.