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?

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?

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

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

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

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

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.

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

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

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?

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

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

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

A Sudoku with clues given as sums of entries.

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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?

Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's there is one digit, between the two 2's there are two digits, and between the two 3's there are three digits.

Find out what a "fault-free" rectangle is and try to make some of your own.

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

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

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?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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

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

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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

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?

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?

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

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

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.

Find out about Magic Squares in this article written for students. Why are they magic?!

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

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

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

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....?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

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

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

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?