We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
How many models can you find which obey these rules?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
What is the largest cuboid you can wrap in an A3 sheet of paper?
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 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?
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.
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.
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 many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Investigate the number of faces you can see when you arrange three cubes in different ways.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
How many triangles can you make on the 3 by 3 pegboard?
Bernard Bagnall describes how to get more out of some favourite
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 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.
Investigate the different ways you could split up these rooms so
that you have double the number.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Why does the tower look a different size in each of these pictures?
In how many ways can you stack these rods, following the rules?
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?
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?
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?
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"?
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?
In my local town there are three supermarkets which each has a
special deal on some products. If you bought all your shopping in
one shop, where would be the cheapest?
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?
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?
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
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
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.
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
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
Follow the directions for circling numbers in the matrix. Add all
the circled numbers together. Note your answer. Try again with a
different starting number. What do you notice?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
If the answer's 2010, what could the question be?
Can you find ways of joining cubes together so that 28 faces are