What shapes can you draw without taking your pen off the paper? (Euler’s Theorem in Graph Theory)

What shapes can you draw without taking your pen off the paper? Does it matter which dot, or vertex, you start from? And what does it all have to do with Euler?

In this brief introduction to graph theory, we explore how you can confidently answer the question “Can you draw this shape without taking your pen off the paper?” no matter what the shape is. Euler’s Theorem tells you which shapes can (and cannot) be drawn without going over the same edge more than once by simply counting the number of lines entering and leaving each vertex.

Produced by Beth Thomas with assistance from Dr Tom Crawford at the University of Oxford.

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