How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
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.
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
An activity making various patterns with 2 x 1 rectangular tiles.
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
In how many ways can you stack these rods, following the rules?
Investigate the different ways you could split up these rooms so
that you have double the number.
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 many models can you find which obey these rules?
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?
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?
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?
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
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!
Sort the houses in my street into different groups. Can you do it in any other ways?
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 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?
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?
How many triangles can you make on the 3 by 3 pegboard?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
Can you find ways of joining cubes together so that 28 faces are
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.
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
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?
Can you create more models that follow these rules?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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?
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?
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
Explore the triangles that can be made with seven sticks of the
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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?
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?
A challenging activity focusing on finding all possible ways of stacking rods.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
This challenge extends the Plants investigation so now four or more children are involved.