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

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

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?

How many models can you find which obey these rules?

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?

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?

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.

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?

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?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

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

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?

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

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?

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

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?

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

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

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

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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?

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

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?

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

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

You could use just coloured pencils and paper to create this design, but it will be more eye-catching if you can get hold of hammer, nails and string.

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

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

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

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

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

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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?

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

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

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

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?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Can you make the birds from the egg tangram?

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?