Powers and roots

We start with one yellow cube and build around it to make a 3×3×3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

Which of these pocket money systems would you rather have?

What is the last digit of the number 1 / 5^903 ?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.

Simple additions can lead to intriguing results...

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay