If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

Use these head, body and leg pieces to make Robot Monsters which are different heights.

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Try out the lottery that is played in a far-away land. What is the chance of winning?

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

Can you use the information to find out which cards I have used?

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

Can you replace the letters with numbers? Is there only one solution in each case?

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

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

Find all the numbers that can be made by adding the dots on two dice.

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

Can you find all the different ways of lining up these Cuisenaire rods?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

These two group activities use mathematical reasoning - one is numerical, one geometric.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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.

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Can you find all the ways to get 15 at the top of this triangle of numbers?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?