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?
Investigate the different ways you could split up these rooms so
that you have double the number.
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!
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?
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.
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.
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
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?
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
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
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?
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?
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?
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?
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 models can you find which obey these rules?
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.
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?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
An investigation that gives you the opportunity to make and justify
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
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?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
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?
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?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?