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?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
Can you find all the different triangles on these peg boards, and
find their angles?
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?
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
This activity investigates how you might make squares and pentominoes from Polydron.
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?
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
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?
Can you draw a square in which the perimeter is numerically equal
to the area?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How many triangles can you make on the 3 by 3 pegboard?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
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?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
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?
A mathematician goes into a supermarket and buys four items. Using
a calculator she multiplies the cost instead of adding them. How
can her answer be the same as the total at the till?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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 smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
The Vikings communicated in writing by making simple scratches on
wood or stones called runes. Can you work out how their code works
using the table of the alphabet?
How many rectangles can you find in this shape? Which ones are
differently sized and which are 'similar'?
Systematically explore the range of symmetric designs that can be
created by shading parts of the motif below. Use normal square
lattice paper to record your results.
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99
How many ways can you do it?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Tim's class collected data about all their pets. Can you put the
animal names under each column in the block graph using the
There are seven pots of plants in a greenhouse. They have lost
their labels. Perhaps you can help re-label them.
Alice and Brian are snails who live on a wall and can only travel
along the cracks. Alice wants to go to see Brian. How far is the
shortest route along the cracks? Is there more than one way to go?
A merchant brings four bars of gold to a jeweller. How can the
jeweller use the scales just twice to identify the lighter, fake
Can you recreate these designs? What are the basic units? What
movement is required between each unit? Some elegant use of
procedures will help - variables not essential.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
Pentagram Pylons - can you elegantly recreate them? Or, the
European flag in LOGO - what poses the greater problem?
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
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!
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?
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?
Investigate the different ways you could split up these rooms so
that you have double the number.