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

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

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?

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?

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

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.

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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?

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

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

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

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?

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?

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

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

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

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

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

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?

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?

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

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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

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

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?

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

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

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

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?

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

How many models can you find which obey these rules?

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

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?

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

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

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.

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?

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

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

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

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 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?

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

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

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