Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Can you use this information to work out Charlie's house number?

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.

Are these statements always true, sometimes true or never true?

The discs for this game are kept in a flat square box with a square hole for each disc. Use the information to find out how many discs of each colour there are in the box.

This activity creates an opportunity to explore all kinds of number-related patterns.

Find another number that is one short of a square number and when you double it and add 1, the result is also a square number.

Can you make square numbers by adding two prime numbers together?

Does a graph of the triangular numbers cross a graph of the six times table? If so, where? Will a graph of the square numbers cross the times table too?

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

In 1871 a mathematician called Augustus De Morgan died. De Morgan made a puzzling statement about his age. Can you discover which year De Morgan was born in?

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

A woman was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. When was she born?

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

Using your knowledge of the properties of numbers, can you fill all the squares on the board?