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
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
A follow-up activity to Tiles in the Garden.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
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?
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
Can you create more models that follow these rules?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
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.
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
This problem is intended to get children to look really hard at something they will see many times in the next few months.
An investigation that gives you the opportunity to make and justify
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.
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.
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
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
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 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.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
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.
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
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.
Can you find ways of joining cubes together so that 28 faces are
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?
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?
If the answer's 2010, what could the question be?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
Investigate these hexagons drawn from different sized equilateral
What is the largest cuboid you can wrap in an A3 sheet of paper?
The red ring is inside the blue ring in this picture. Can you
rearrange the rings in different ways? Perhaps you can overlap them
or put one outside another?
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?