Lost in Three Dimensions

Pólya’s theorem

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Random walks in 1D and 2D return to the start with probability 1.
• In 3D the return probability is only about 34%.
• “A drunk man will find his way home, but a drunk bird may get lost forever.”

Steps

1. Will you get back?
2. Two steps and back
3. Counting returns
4. From returns to probability
5. Back home on a grid
6. Drunk people and birds