Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
If I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
An activity making various patterns with 2 x 1 rectangular tiles.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
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?
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?
Here are many ideas for you to investigate - all linked with the
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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 do these two triangles have in common? How are they related?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Have a go at this 3D extension to the Pebbles problem.
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?
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
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.
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?
Investigate the number of faces you can see when you arrange three cubes in different ways.
How many tiles do we need to tile these patios?
An investigation that gives you the opportunity to make and justify
A follow-up activity to Tiles in the Garden.
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
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.
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 how this pattern of squares continues. You could
measure lengths, areas and angles.
Explore one of these five pictures.
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
What is the largest cuboid you can wrap in an A3 sheet of paper?
In my local town there are three supermarkets which each has a
special deal on some products. If you bought all your shopping in
one shop, where would be the cheapest?
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?
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!
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Investigate the different ways you could split up these rooms so
that you have double the number.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
Can you create more models that follow these rules?
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
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.