Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
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?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
In how many ways can you stack these rods, following the rules?
Explore the triangles that can be made with seven sticks of the
Can you create more models that follow these rules?
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
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?
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?
These pictures show squares split into halves. Can you find other ways?
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?
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.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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?
How many models can you find which obey these rules?
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Can you find ways of joining cubes together so that 28 faces are
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
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
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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
How many triangles can you make on the 3 by 3 pegboard?
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.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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?
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
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!