Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
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?
How many triangles can you make on the 3 by 3 pegboard?
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?
These pictures show squares split into halves. Can you find other
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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.
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.
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
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.
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?
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?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
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?
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?
Can you create more models that follow these rules?
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?
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
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
How many models can you find which obey these rules?
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?
In how many ways can you stack these rods, following the rules?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
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?
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?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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.
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?
Explore the triangles that can be made with seven sticks of the
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Can you find ways of joining cubes together so that 28 faces are
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 Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.