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?
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
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?
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.
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?
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!
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
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.
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
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 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?
In how many ways can you stack these rods, following the rules?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
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?
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?
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?
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?
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.
An activity making various patterns with 2 x 1 rectangular tiles.
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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.
If the answer's 2010, what could the question be?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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 cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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?
Well now, what would happen if we lost all the nines in our number
system? Have a go at writing the numbers out in this way and have a
look at the multiplications table.
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
An investigation that gives you the opportunity to make and justify
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
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?
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?
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.