Ben’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?

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

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?

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?

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

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

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

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

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. What happens?

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?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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.

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

Try out this number trick. What happens with different starting numbers? What do you notice?

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

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

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?

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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

Here are two kinds of spirals for you to explore. What do you notice?

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

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

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

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.

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

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?

Find the sum of all three-digit numbers each of whose digits is odd.

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?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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?

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.

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?

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?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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

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?

This challenge asks you to imagine a snake coiling on itself.

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

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