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?

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.

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?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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?

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

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?

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?

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

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?

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

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

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

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.

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

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

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.

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?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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?

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....?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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?

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

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.

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

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

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

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

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?

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

How many models can you find which obey these rules?

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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