This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

An investigation that gives you the opportunity to make and justify predictions.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

This challenge is about finding the difference between numbers which have the same tens digit.

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

This task follows on from Build it Up and takes the ideas into three dimensions!

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

How many centimetres of rope will I need to make another mat just like the one I have here?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

What happens when you round these three-digit numbers to the nearest 100?

Got It game for an adult and child. How can you play so that you know you will always win?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these numbers to the nearest whole number?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Find out what a "fault-free" rectangle is and try to make some of your own.

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?