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?
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?
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?
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.
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
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.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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
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?
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?
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?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
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?
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?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Can you put the 25 coloured tiles into the 5 x 5 square so that no
column, no row and no diagonal line have tiles of the same colour
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
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 different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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 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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
How many different symmetrical shapes can you make by shading triangles or squares?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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
How many models can you find which obey these rules?
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.
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
What could the half time scores have been in these Olympic hockey
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!