Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
What is the largest cuboid you can wrap in an A3 sheet of paper?
We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
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.
Investigate the number of faces you can see when you arrange three cubes in different ways.
While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Can you create more models that follow these rules?
How many models can you find which obey these rules?
What do these two triangles have in common? How are they related?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
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?
Have a go at this 3D extension to the Pebbles problem.
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
A description of some experiments in which you can make discoveries about triangles.
I cut this square into two different shapes. What can you say about the relationship between them?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
An investigation that gives you the opportunity to make and justify predictions.
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
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?
I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?
An activity making various patterns with 2 x 1 rectangular tiles.
Investigate what happens when you add house numbers along a street in different ways.
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?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Why does the tower look a different size in each of these pictures?
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
Explore one of these five pictures.
How many tiles do we need to tile these patios?
In how many ways can you stack these rods, following the rules?