Don't like trigonometry? Don't worry, here are three beautiful proofs of a well-known result that make do without it.

Imagine a circle with radius 1 cm rolling completely along the circumference of a circle with radius 4 cm. How many rotations did the smaller circle make? Be prepared for a surprise!

The number 1 can be written as a sum of unit fractions, that is fractions with 1 in the numerator. But how long can we make such a sum?