Creating and manipulating expressions and formulae

Surprising numerical patterns can be explained using algebra and diagrams...

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

Which armies can be arranged in hollow square fighting formations?

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

What's special about the area of quadrilaterals drawn in a square?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Can you fit polynomials through these points?

By proving these particular identities, prove the existence of general cases.