If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

You have a set of the digits from 0 – 9. Can you arrange these in the 5 boxes to make two-digit numbers as close to the targets as possible?

Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.

Can you each work out the number on your card? What do you notice? How could you sort the cards?

Can you each work out what shape you have part of on your card? What will the rest of it look like?

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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 split each of the shapes below in half so that the two parts are exactly the same?

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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

These pictures show squares split into halves. Can you find other ways?

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?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

You will need a long strip of paper for this task. Cut it into different lengths. How could you find out how long each piece is?

Here is a version of the game 'Happy Families' for you to make and play.

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

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?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

Explore the triangles that can be made with seven sticks of the same length.

Can you put these shapes in order of size? Start with the smallest.

Can you deduce the pattern that has been used to lay out these bottle tops?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

Can you see which tile is the odd one out in this design? Using the basic tile, can you make a repeating pattern to decorate our wall?

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 make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

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

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

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Can you make five differently sized squares from the tangram pieces?

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

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?

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?

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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?

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?

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.