This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
Can you create more models that follow these rules?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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?
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
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
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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.
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.
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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 models can you find which obey these rules?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Have a go at this 3D extension to the Pebbles problem.
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?
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.
In how many ways can you stack these rods, following the rules?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
A challenging activity focusing on finding all possible ways of stacking rods.
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.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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!
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
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.
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
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?
How many triangles can you make on the 3 by 3 pegboard?
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?