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.
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?
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?
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?
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?
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 many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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?
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?
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?
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?
How many models can you find which obey these rules?
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Can you cut up a square in the way shown and make the pieces into a triangle?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
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?
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Here's a simple way to make a Tangram without any measuring or ruling lines.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
This activity investigates how you might make squares and pentominoes from Polydron.
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.
Can you create more models that follow these rules?
Delight your friends with this cunning trick! Can you explain how it works?
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?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
Can you each work out the number on your card? What do you notice? How could you sort the cards?
What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
Can you visualise what shape this piece of paper will make when it is folded?
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?
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 ...
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?