Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

It would be nice to have a strategy for disentangling any tangled ropes...

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Is it possible to find the angles in this rather special isosceles triangle?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Which numbers can we write as a sum of square numbers?

Can you create a Latin Square from multiples of a six digit number?