Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
How many triangles can you make on the 3 by 3 pegboard?
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?
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?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
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 do these two triangles have in common? How are they related?
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?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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.
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?
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 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.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
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.
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?
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?
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 many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
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 candle and sundial?
It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!
Can you fit the tangram pieces into the outline of these rabbits?
Can you fit the tangram pieces into the outlines of the workmen?
Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?
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 outlines of Mai Ling and Chi Wing?
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 goat and giraffe?
Can you fit the tangram pieces into the outline of this plaque design?
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 shape. How would you describe it?
Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?
Can you logically construct these silhouettes using the tangram pieces?
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?
Follow the diagrams to make this patchwork piece, based on an octagon in a square.
Ideas for practical ways of representing data such as Venn and Carroll diagrams.
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 Wai Ping, Wah Ming and Chi Wing?