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.

How many models can you find which obey these rules?

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

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?

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?

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?

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

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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?

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

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?

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

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?

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.

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?

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.

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

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?

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

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

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

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

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?

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?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Can you fit the tangram pieces into the outlines of the candle and sundial?

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outline of this plaque design?

Here's a simple way to make a Tangram without any measuring or ruling lines.

Can you make the birds from the egg tangram?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outline of this goat and giraffe?

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

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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

Can you fit the tangram pieces into the outline of this telephone?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

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?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?