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?
How many models can you find which obey these rules?
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?
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 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.
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?
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?
How many triangles can you make on the 3 by 3 pegboard?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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.
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?
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?
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
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?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
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 create more models that follow these rules?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
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 Little Fung at the table?
Can you fit the tangram pieces into the outline of Little Ming playing the board game?
You'll need a collection of cups for this activity.
For this activity which explores capacity, you will need to collect some bottles and jars.
Can you fit the tangram pieces into the outline of this telephone?
Can you fit the tangram pieces into the outline of the child walking home from school?
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...
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?
Can you fit the tangram pieces into the outlines of the candle and sundial?
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 chairs?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
In this activity focusing on capacity, you will need a collection of different jars and bottles.