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?
Investigate the number of faces you can see when you arrange three cubes in different ways.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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 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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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?
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?
How many models can you find which obey these rules?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
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?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
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?
Why does the tower look a different size in each of these pictures?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.
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?
Here are many ideas for you to investigate - all linked with the number 2000.
Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
How many tiles do we need to tile these patios?
In how many ways can you stack these rods, following the rules?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?
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?
Have a go at this 3D extension to the Pebbles problem.
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
What do these two triangles have in common? How are they related?
Can you create more models that follow these rules?
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.
An investigation that gives you the opportunity to make and justify predictions.
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?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?