The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.

A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis. How many paths are there for 4 steps, for 6 steps, for 8 steps?

Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.