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?
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?
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.
How many triangles can you make on the 3 by 3 pegboard?
Investigate these hexagons drawn from different sized equilateral
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?
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.
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
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 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?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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!
An activity making various patterns with 2 x 1 rectangular tiles.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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
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
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 I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
Investigate what happens when you add house numbers along a street
in different ways.
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 investigation that gives you the opportunity to make and justify
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?
Investigate this balance which is marked in halves. If you had a
weight on the left-hand 7, where could you hang two weights on the
right to make it balance?
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
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?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Investigate the number of faces you can see when you arrange three cubes in different ways.
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
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?
In my local town there are three supermarkets which each has a
special deal on some products. If you bought all your shopping in
one shop, where would be the cheapest?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
How many tiles do we need to tile these patios?
Why does the tower look a different size in each of these pictures?
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?