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?
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.
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.
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 we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
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?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
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?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
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.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
How many triangles can you make on the 3 by 3 pegboard?
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?
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!
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
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?
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 how many ways can you stack these rods, following the rules?
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 different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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
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?
Why does the tower look a different size in each of these pictures?
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 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?
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?
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 a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Sort the houses in my street into different groups. Can you do it in any other ways?
The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?
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
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?
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?