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.

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?

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?

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

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

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.

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?

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.

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

How many trains can you make which are the same length as Matt's, using rods that are identical?

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?

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

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

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

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.

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

How many models can you find which obey these rules?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

How many different triangles can you make on a circular pegboard that has nine pegs?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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?

Can you draw a square in which the perimeter is numerically equal to the area?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

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

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

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?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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

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

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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.

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

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

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?

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

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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?