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?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
Can you create more models that follow these rules?
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.
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Can you find ways of joining cubes together so that 28 faces are
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
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?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
Have a go at this 3D extension to the Pebbles problem.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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?
How many models can you find which obey these rules?
An activity making various patterns with 2 x 1 rectangular tiles.
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 do these two triangles have in common? How are they related?
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?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
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?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
In how many ways can you stack these rods, following the rules?
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
This challenge extends the Plants investigation so now four or more children are involved.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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 continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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!
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?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Investigate the different ways you could split up these rooms so
that you have double the number.
Investigate the number of faces you can see when you arrange three cubes in different 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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
How many triangles can you make on the 3 by 3 pegboard?
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?