Creating and manipulating expressions and formulae

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Label this plum tree graph to make it totally magic!

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying them. Can you explain how they work?

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Can you work out the value of x in this 'power-full' equation?

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?