How many triangles can you make on the 3 by 3 pegboard?
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?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
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?
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?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
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?
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.
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 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?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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 split each of the shapes below in half so that the two parts are exactly the same?
This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?
These pictures show squares split into halves. Can you find other ways?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
These practical challenges are all about making a 'tray' and covering it with paper.
Can you describe a piece of paper clearly enough for your partner to know which piece it is?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
Explore the triangles that can be made with seven sticks of the same length.
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
Our 2008 Advent Calendar has a 'Making Maths' activity for every 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 outlines of Mai Ling and Chi Wing?
Can you fit the tangram pieces into the outlines of the candle and sundial?
Can you fit the tangram pieces into the outlines of the workmen?
Can you fit the tangram pieces into the outline of these rabbits?
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 this shape. How would you describe it?
Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
Can you fit the tangram pieces into the outline of this telephone?
Can you see which tile is the odd one out in this design? Using the basic tile, can you make a repeating pattern to decorate our wall?
Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?