Expanding and factorising quadratics

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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

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?

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

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Can you produce convincing arguments that a selection of statements about numbers are true?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.