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

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?

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?

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?

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.

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.

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

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?

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.

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?

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?

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

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.

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

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

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

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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

How many models can you find which obey these rules?

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

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?

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

Can you make the birds from the egg tangram?

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

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

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

What do these two triangles have in common? How are they related?

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

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

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

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

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 cube out of straws and have a go at this practical challenge.

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?