Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

These practical challenges are all about making a 'tray' and covering it with paper.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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.

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

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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

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?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

What is the best way to shunt these carriages so that each train can continue its journey?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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

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

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?

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.

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

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

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

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.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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

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

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

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.

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

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

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Find your way through the grid starting at 2 and following these operations. What number do you end on?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

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?

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?

What happens when you try and fit the triomino pieces into these two grids?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?

An activity making various patterns with 2 x 1 rectangular tiles.

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?