What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
What is the largest cuboid you can wrap in an A3 sheet of paper?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
Investigate these hexagons drawn from different sized equilateral
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
Ben has five coins in his pocket. How much money might he have?
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.
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.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
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
An investigation that gives you the opportunity to make and justify
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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
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?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Why does the tower look a different size in each of these pictures?
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?
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?
How many tiles do we need to tile these patios?
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?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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.
"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.
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
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
We went to the cinema and decided to buy some bags of popcorn so we
asked about the prices. Investigate how much popcorn each bag holds
so find out which we might have bought.
An activity making various patterns with 2 x 1 rectangular tiles.
Investigate what happens when you add house numbers along a street
in different ways.
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!
In this investigation, you must try to make houses using cubes. If
the base must not spill over 4 squares and you have 7 cubes which
stand for 7 rooms, what different designs can you come up with?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
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?
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?
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?
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.
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.
Complete these two jigsaws then put one on top of the other. What
happens when you add the 'touching' numbers? What happens when you
change the position of the jigsaws?
How many models can you find which obey these rules?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.