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?
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 we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
How many models can you find which obey these rules?
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
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.
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?
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?
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
These practical challenges are all about making a 'tray' and covering it with paper.
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 can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Can you shunt the trucks so that the Cattle truck and the Sheep
truck change places and the Engine is back on the main line?
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.
What is the best way to shunt these carriages so that each train
can continue its journey?
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?
In how many ways can you stack these rods, following the rules?
Can you find all the different ways of lining up these Cuisenaire
Put 10 counters in a row. Find a way to arrange the counters into
five pairs, evenly spaced in a row, in just 5 moves, using the
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.
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
How many triangles can you make on the 3 by 3 pegboard?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
Can you draw a square in which the perimeter is numerically equal
to the area?
If you had 36 cubes, what different cuboids could you make?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
An investigation that gives you the opportunity to make and justify
Design an arrangement of display boards in the school hall which fits the requirements of different people.
These two group activities use mathematical reasoning - one is
numerical, one geometric.
Your challenge is to find the longest way through the network
following this rule. You can start and finish anywhere, and with
any shape, as long as you follow the correct order.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
When intergalactic Wag Worms are born they look just like a cube.
Each year they grow another cube in any direction. Find all the
shapes that five-year-old Wag Worms can be.
Investigate the different ways you could split up these rooms so
that you have double the number.
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
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?
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?
Penta people, the Pentominoes, always build their houses from five
square rooms. I wonder how many different Penta homes you can