This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
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?
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?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
Place the 16 different combinations of cup/saucer in this 4 by 4
arrangement so that no row or column contains more than one cup or
saucer of the same colour.
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
In how many ways can you stack these rods, following the rules?
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?
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!
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?
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
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?
There are to be 6 homes built on a new development site. They could
be semi-detached, detached or terraced houses. How many different
combinations of these can you find?
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.
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?
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?
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
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
In this investigation, you are challenged to make mobile phone
numbers which are easy to remember. What happens if you make a
sequence adding 2 each time?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
An activity making various patterns with 2 x 1 rectangular tiles.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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?
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?
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?
This challenge extends the Plants investigation so now four or more children are involved.
How many models can you find which obey these rules?
Can you create more models that follow these rules?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?