What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
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?
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.
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?
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 triangles can you make on the 3 by 3 pegboard?
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.
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?
This activity investigates how you might make squares and pentominoes from Polydron.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
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?
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?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
An activity making various patterns with 2 x 1 rectangular tiles.
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
Can you create more models that follow these rules?
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?
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?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Can you fit the tangram pieces into the outline of this plaque design?
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you fit the tangram pieces into the outline of the telescope and microscope?
Exploring and predicting folding, cutting and punching holes and making spirals.
Make a cube out of straws and have a go at this practical challenge.
Did you know mazes tell stories? Find out more about mazes and make one of your own.
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 outline of Little Ming and Little Fung dancing?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
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 Wai Ping, Wah Ming and Chi Wing?
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?
Can you fit the tangram pieces into the outline of Little Fung at the table?
Can you fit the tangram pieces into the outlines of these people?
Can you fit the tangram pieces into the outlines of the workmen?
Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.
Can you fit the tangram pieces into the outline of the child walking home from school?