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?

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

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?

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?

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?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

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

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

In how many ways can you stack these rods, following the rules?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

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!

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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.

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.

There are lots of different methods to find out what the shapes are worth - how many can you find?

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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

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!

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

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Can you substitute numbers for the letters in these sums?

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. . . .

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Investigate the different ways you could split up these rooms so that you have double the number.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.