These practical challenges are all about making a 'tray' and covering it with paper.
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 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?
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
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?
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
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 shunt the trucks so that the Cattle truck and the Sheep
truck change places and the Engine is back on the main line?
This problem focuses on Dienes' Logiblocs. What is the same and
what is different about these pairs of shapes? Can you describe the
shapes in the picture?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
What is the best way to shunt these carriages so that each train
can continue its journey?
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
Investigate the different ways you could split up these rooms so
that you have double the number.
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
In how many ways could Mrs Beeswax put ten coins into her three
puddings so that each pudding ended up with at least two coins?
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
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?
Exactly 195 digits have been used to number the pages in a book.
How many pages does the book have?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.
How many models can you find which obey these rules?
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?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Can you find all the different ways of lining up these Cuisenaire
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
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?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
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?
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.
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?
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.
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
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....?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?
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
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. . . .