Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
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?
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.
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 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.
These practical challenges are all about making a 'tray' and covering it with paper.
How many models can you find which obey these rules?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
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?
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?
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?
How many triangles can you make on the 3 by 3 pegboard?
What do these two triangles have in common? How are they related?
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?
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?
Can you create more models that follow these rules?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
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?
This activity investigates how you might make squares and pentominoes from Polydron.
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?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
How do you know if your set of dominoes is complete?
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 practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Can you fit the tangram pieces into the outline of Little Ming playing the board game?
Can you fit the tangram pieces into the outline of Little Fung at the table?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
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?
Delight your friends with this cunning trick! Can you explain how it works?
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?
Can you fit the tangram pieces into the outlines of these people?
Can you fit the tangram pieces into the outlines of these clocks?
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 outlines of Mai Ling and Chi Wing?
Can you fit the tangram pieces into the outlines of the workmen?
Can you fit the tangram pieces into the outlines of the chairs?