How many tiles do we need to tile these patios?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Here are many ideas for you to investigate - all linked with the
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
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?
Bernard Bagnall describes how to get more out of some favourite
Investigate these hexagons drawn from different sized equilateral
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
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?
Why does the tower look a different size in each of these pictures?
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
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?
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 and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
What is the largest cuboid you can wrap in an A3 sheet of paper?
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?
How many different sets of numbers with at least four members can
you find in the numbers in this box?
Investigate the different ways these aliens count in this
challenge. You could start by thinking about how each of them would
write our number 7.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
In this section from a calendar, put a square box around the 1st,
2nd, 8th and 9th. Add all the pairs of numbers. What do you notice
about the answers?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
This activity asks you to collect information about the birds you
see in the garden. Are there patterns in the data or do the birds
seem to visit randomly?
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.
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
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
Investigate the number of faces you can see when you arrange three cubes in different ways.
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
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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?
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 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?
I cut this square into two different shapes. What can you say about
the relationship between them?
If the answer's 2010, what could the question be?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
How many models can you find which obey these rules?
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?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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?
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.
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?
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.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?