A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?

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?

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 make square numbers by adding two prime numbers together?

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.

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.

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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?

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

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

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?

Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?

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

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

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?

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.

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?

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

Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.

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

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?

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?

A square patio was tiled with square tiles all the same size. Some of the tiles were removed from the middle of the patio in order to make a square flower bed, but the number of the remaining tiles. . . .

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Look at the squares in this problem. What does the next square look like? I draw a square with 81 little squares inside it. How long and how wide is my square?

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