Place this "worm" on the 100 square and find the total of the four
squares it covers. Keeping its head in the same place, what other
totals can you make?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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!
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
How many triangles can you make on the 3 by 3 pegboard?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
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?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
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?
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?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
How many models can you find which obey these rules?
In how many ways can you stack these rods, following the rules?
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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?
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 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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
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?
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 different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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
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?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Ben has five coins in his pocket. How much money might he have?
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.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Investigate what happens when you add house numbers along a street
in different ways.
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
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
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?