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
Can you draw a square in which the perimeter is numerically equal
to the area?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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?
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.
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?
This activity investigates how you might make squares and pentominoes from Polydron.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.
My cube has inky marks on each face. Can you find the route it has
taken? What does each face look like?
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?
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.
How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?
Moira is late for school. What is the shortest route she can take from the school gates to the entrance?
My coat has three buttons. How many ways can you find to do up all
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 ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Imagine that the puzzle pieces of a jigsaw are roughly a
rectangular shape and all the same size. How many different puzzle
pieces could there be?
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?
What could the half time scores have been in these Olympic hockey
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
When intergalactic Wag Worms are born they look just like a cube.
Each year they grow another cube in any direction. Find all the
shapes that five-year-old Wag Worms can be.
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?
These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Sally and Ben were drawing shapes in chalk on the school
playground. Can you work out what shapes each of them drew using
Can you rearrange the biscuits on the plates so that the three
biscuits on each plate are all different and there is no plate with
two biscuits the same as two biscuits on another plate?
In the planet system of Octa the planets are arranged in the shape
of an octahedron. How many different routes could be taken to get
from Planet A to Planet Zargon?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Find the product of the numbers on the routes from A to B. Which
route has the smallest product? Which the largest?
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.
How many models can you find which obey these rules?
The planet of Vuvv has seven moons. Can you work out how long it is
between each super-eclipse?
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
An investigation that gives you the opportunity to make and justify