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, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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?

Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?

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?

You have 5 darts and your target score is 44. How many different ways could you score 44?

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.

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?

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

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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!

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!

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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?

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

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?

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

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

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

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.

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

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

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.

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?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

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

Fill in the missing numbers so that adding each pair of corner numbers gives you the number between them (in the box).

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total.

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?

Place the digits 1 to 9 into the circles so that each side of the triangle adds to the same total.

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

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

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?

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

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?

Investigate the different distances of these car journeys and find out how long they take.

Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

Find a great variety of ways of asking questions which make 8.