In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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"?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
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?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Why does the tower look a different size in each of these pictures?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
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
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 ways can you find of tiling the square patio, using square
tiles of different sizes?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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.
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
Investigate the number of faces you can see when you arrange three cubes in different ways.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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
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?
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?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
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?
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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.
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?
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.
If the answer's 2010, what could the question be?
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?
Can you find ways of joining cubes together so that 28 faces are
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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 out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
Ben has five coins in his pocket. How much money might he have?
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?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?