10-Second Challenge

10-Second Challenge #1

Can you draw each graph without lifting your pencil?

Try both graphs before looking at the solution.

Show the solution

Graph A can be drawn in one continuous stroke without lifting your pencil or retracing any line.

One possible route is:

1 → 2 → 3 → 1 → 4 → 2 → 5 → 3 → 4 → 5 → 1

This route uses every edge exactly once and returns to the starting point.

Count the number of edges connected to each vertex:

Every vertex has an even degree.

That means Graph A has an Euler circuit: a route that uses every edge exactly once and ends where it started.

Graph B cannot be drawn in one continuous stroke without retracing an edge.

Count the edges connected to each vertex:

Vertices 1, 2, 3, and 4 all have odd degree.

A connected graph can be drawn in one continuous stroke only when it has:

0 odd-degree vertices — you can start and finish at the same vertex, or
2 odd-degree vertices — you start at one odd vertex and finish at the other.

Graph B has 4 odd-degree vertices, so it cannot be drawn in one continuous stroke.

This puzzle is an example of graph theory.

A graph is made of:

The number of edges meeting at a vertex is called its degree.

This simple idea lets you determine whether a drawing can be completed in one stroke — without trying every possible route.