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?
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 ways can you find of tiling the square patio, using square
tiles of different sizes?
Investigate the different ways you could split up these rooms so
that you have double the number.
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
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?
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?
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?
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?
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?
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.
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?
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.
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?
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
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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!
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.
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?
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?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
How many models can you find which obey these rules?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
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.
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 triangles can you make on the 3 by 3 pegboard?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
An investigation that gives you the opportunity to make and justify
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Ben has five coins in his pocket. How much money might he have?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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?
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?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a