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?
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?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
Have a go at this 3D extension to the Pebbles problem.
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Can you create more models that follow these rules?
Explore the triangles that can be made with seven sticks of the same length.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What do these two triangles have in common? How are they related?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
Can you find ways of joining cubes together so that 28 faces are visible?
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
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.
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
These pictures show squares split into halves. Can you find other ways?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
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.
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?
How many triangles can you make on the 3 by 3 pegboard?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
How many models can you find which obey these rules?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
An activity making various patterns with 2 x 1 rectangular tiles.
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
Why does the tower look a different size in each of these pictures?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?
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.
Explore one of these five pictures.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?
If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?
How many tiles do we need to tile these patios?