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.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
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 triangles can you make on the 3 by 3 pegboard?
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?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
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?
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?
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.
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Factors and Multiples game for an adult and child. How can you make sure you win this game?
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.
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
Can you each work out the number on your card? What do you notice? How could you sort the cards?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
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?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
This activity investigates how you might make squares and pentominoes from Polydron.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Can you create more models that follow these rules?
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 outline of this telephone?
Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?
Can you fit the tangram pieces into the outline of Granma T?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
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 outlines of these clocks?
What do these two triangles have in common? How are they related?
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 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 outlines of the lobster, yacht and cyclist?