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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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
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
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?
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.
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
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?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
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
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
Moira is late for school. What is the shortest route she can take from the school gates to the entrance?
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?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
My coat has three buttons. How many ways can you find to do up all
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
El Crico the cricket has to cross a square patio to get home. He
can jump the length of one tile, two tiles and three tiles. Can you
find a path that would get El Crico home in three jumps?
Can you find out in which order the children are standing in this
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
Is it possible to place 2 counters on the 3 by 3 grid so that there
is an even number of counters in every row and every column? How
about if you have 3 counters or 4 counters or....?
Can you cover the camel with these pieces?
Use the information to describe these marbles. What colours must be
on marbles that sparkle when rolling but are dark inside?
What happens when you try and fit the triomino pieces into these
How many models can you find which obey these rules?
Find all the different shapes that can be made by joining five
equilateral triangles edge to edge.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Start with three pairs of socks. Now mix them up so that no
mismatched pair is the same as another mismatched pair. Is there
more than one way to do it?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
Investigate the different ways you could split up these rooms so
that you have double the number.
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
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?
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?
An investigation that gives you the opportunity to make and justify
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?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Can you find all the different ways of lining up these Cuisenaire
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
In how many ways can you stack these rods, following the rules?