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
These practical challenges are all about making a 'tray' and covering it with paper.
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
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 we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
How many models can you find which obey these rules?
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?
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 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 different triangles can you make on a circular pegboard that has nine pegs?
Can you draw a square in which the perimeter is numerically equal
to the area?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
What is the best way to shunt these carriages so that each train
can continue its journey?
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
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 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?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
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....?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
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?
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How many trains can you make which are the same length as Matt's, using rods that are identical?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
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?
Use the information to describe these marbles. What colours must be
on marbles that sparkle when rolling but are dark inside?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Can you work out how to balance this equaliser? You can put more
than one weight on a hook.
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
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?
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?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
An investigation that gives you the opportunity to make and justify
Can you find all the different ways of lining up these Cuisenaire
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?