Creating and manipulating expressions and formulae

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

What is special about the difference between squares of numbers adjacent to multiples of three?

If you know the perimeter of a right angled triangle, what can you say about the area?

Can you explain what is going on in these puzzling number tricks?

There are unexpected discoveries to be made about square numbers...

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

A 2-digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Can you use the diagram to prove the AM-GM inequality?