What do these two triangles have in common? How are they related?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
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?
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?
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 is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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.
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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?
This article for teachers suggests ideas for activities built around 10 and 2010.
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?
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?
How many models can you find which obey these rules?
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?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
In how many ways can you stack these rods, following the rules?
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.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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.
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
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.
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!
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
I cut this square into two different shapes. What can you say about
the relationship between them?
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
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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 challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
In this investigation, you must try to make houses using cubes. If
the base must not spill over 4 squares and you have 7 cubes which
stand for 7 rooms, what different designs can you come up with?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
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 create more models that follow these rules?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?