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.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
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?
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 the birds from the egg tangram?
This activity investigates how you might make squares and pentominoes from Polydron.
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?
How many triangles can you make on the 3 by 3 pegboard?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
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?
How many models can you find which obey these rules?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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.
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?
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.
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 practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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.
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
Delight your friends with this cunning trick! Can you explain how it works?
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?
Here is a version of the game 'Happy Families' for you to make and play.
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
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.
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 create more models that follow these rules?
Here's a simple way to make a Tangram without any measuring or ruling lines.
Can you fit the tangram pieces into the outline of this telephone?
Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?
Can you fit the tangram pieces into the outline of Little Ming playing the board game?
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?
Make a cube out of straws and have a go at this practical challenge.
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Ideas for practical ways of representing data such as Venn and Carroll diagrams.
Can you fit the tangram pieces into the outline of Granma T?
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?
Exploring and predicting folding, cutting and punching holes and making spirals.
Can you fit the tangram pieces into the outline of this junk?
The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.