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?
Investigate the number of faces you can see when you arrange three cubes in different ways.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
What do these two triangles have in common? How are they related?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
A follow-up activity to Tiles in the Garden.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Bernard Bagnall describes how to get more out of some favourite
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
An investigation that gives you the opportunity to make and justify
Can you create more models that follow these rules?
How many models can you find which obey these rules?
Explore one of these five pictures.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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?
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?
Have a go at this 3D extension to the Pebbles problem.
This problem is intended to get children to look really hard at something they will see many times in the next few months.
Why does the tower look a different size in each of these pictures?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 how this pattern of squares continues. You could
measure lengths, areas and angles.
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
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
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?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
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?
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
Explore Alex's number plumber. What questions would you like to
ask? Don't forget to keep visiting NRICH projects site for the
latest developments and questions.
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?
If the answer's 2010, what could the question be?
Can you find ways of joining cubes together so that 28 faces are
I cut this square into two different shapes. What can you say about
the relationship between them?
Lolla bought a balloon at the circus. She gave the clown six coins
to pay for it. What could Lolla have paid for the balloon?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
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?