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
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?
These pictures show squares split into halves. Can you find other ways?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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
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.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
What do these two triangles have in common? How are they related?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
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?
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
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?
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?
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.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
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?
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
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.
What happens when you add the digits of a number then multiply the
result by 2 and you keep doing this? You could try for different
numbers and different rules.
How many models can you find which obey these rules?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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 how many ways can you stack these rods, following the rules?
How many tiles do we need to tile these patios?
Have a go at this 3D extension to the Pebbles problem.
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
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?
An investigation that gives you the opportunity to make and justify
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular