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 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?
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 is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
This activity investigates how you might make squares and pentominoes from Polydron.
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.
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?
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?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
An activity making various patterns with 2 x 1 rectangular tiles.
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
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?
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 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?
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.
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
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.
Can you fit the tangram pieces into the outline of these rabbits?
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 Little Ming and Little Fung dancing?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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 shape. How would you describe it?
Can you fit the tangram pieces into the outline of this plaque design?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
How do you know if your set of dominoes is complete?
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
Can you fit the tangram pieces into the outlines of the chairs?
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?
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
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 logically construct these silhouettes using the tangram pieces?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
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?
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 Little Ming playing the board game?