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?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Can you find the hidden factors which multiply together to produce each quadratic expression?

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

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

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

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

Which armies can be arranged in hollow square fighting formations?

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

A collection of short Stage 4 problems on expanding and factorising quadratics.