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

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

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

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

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

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.

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?

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?

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?

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

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.

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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?

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

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

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.

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

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

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.?

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

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

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?

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

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

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?

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?

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

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

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

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

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.

Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be?

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

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?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

How many models can you find which obey these rules?

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?

My coat has three buttons. How many ways can you find to do up all the buttons?

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

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?

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

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

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