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?
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
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.
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?
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
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.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
How many models can you find which obey these rules?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
An activity making various patterns with 2 x 1 rectangular tiles.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
What do these two triangles have in common? How are they related?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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!
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?
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
These pictures were made by starting with a square, finding the
half-way point on each side and joining those points up. You could
investigate your own starting shape.
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.
Can you create more models that follow these rules?
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?
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?
In how many ways can you stack these rods, following the rules?
How many tiles do we need to tile these patios?
Why does the tower look a different size in each of these pictures?
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
I cut this square into two different shapes. What can you say about
the relationship between them?
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?