How many triangles can you make on the 3 by 3 pegboard?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
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.
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
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
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?
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.
An activity making various patterns with 2 x 1 rectangular tiles.
How many models can you find which obey these rules?
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.
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.
In how many ways can you stack these rods, following the rules?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
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
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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?
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?
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.
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?
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
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?
Explore the triangles that can be made with seven sticks of the
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
Can you find ways of joining cubes together so that 28 faces are
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?
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.
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?
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?
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!