What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
10 space travellers are waiting to board their spaceships. There
are two rows of seats in the waiting room. Using the rules, where
are they all sitting? Can you find all the possible ways?
What is the best way to shunt these carriages so that each train
can continue its journey?
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?
Take a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
These practical challenges are all about making a 'tray' and covering it with paper.
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
An activity making various patterns with 2 x 1 rectangular tiles.
A magician took a suit of thirteen cards and held them in his hand
face down. Every card he revealed had the same value as the one he
had just finished spelling. How did this work?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
In how many ways can you fit two of these yellow triangles
together? Can you predict the number of ways two blue triangles can
be fitted together?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
A tetromino is made up of four squares joined edge to edge. Can
this tetromino, together with 15 copies of itself, be used to cover
an eight by eight chessboard?
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.
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 listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
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?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Starting with four different triangles, imagine you have an
unlimited number of each type. How many different tetrahedra can
you make? Convince us you have found them all.
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.
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
You have 4 red and 5 blue counters. How many ways can they be
placed on a 3 by 3 grid so that all the rows columns and diagonals
have an even number of red counters?
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?
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 we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
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 models can you find which obey these rules?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
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
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.
There are 44 people coming to a dinner party. There are 15 square
tables that seat 4 people. Find a way to seat the 44 people using
all 15 tables, with no empty places.
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
This activity investigates how you might make squares and pentominoes from Polydron.
In how many ways can you stack these rods, following the rules?