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

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

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

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?

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?

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?

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.

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

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?

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

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

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

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?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

This activity investigates how you might make squares and pentominoes from Polydron.

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?

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.

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

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?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

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

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 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.

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

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

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.

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

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

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

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

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.

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?

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

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

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?

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

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?

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