Can you draw a square in which the perimeter is numerically equal
to the area?
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?
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
An investigation that gives you the opportunity to make and justify
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
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
How many different triangles can you make on a circular pegboard that has nine pegs?
Sally and Ben were drawing shapes in chalk on the school
playground. Can you work out what shapes each of them drew using
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
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?
In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?
Can you find all the different triangles on these peg boards, and
find their angles?
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
These practical challenges are all about making a 'tray' and covering it with paper.
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
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
How many possible necklaces can you find? And how do you know you've found them all?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
How many rectangles can you find in this shape? Which ones are
differently sized and which are 'similar'?
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?
How many triangles can you make on the 3 by 3 pegboard?
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?
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?
The puzzle can be solved with the help of small clue-numbers which
are either placed on the border lines between selected pairs of
neighbouring squares of the grid or placed after slash marks on. . . .
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?
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?
Alice's mum needs to go to each child's house just once and then
back home again. How many different routes are there? Use the
information to find out how long each road is on the route she
Can you coach your rowing eight to win?
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....?
Find out about Magic Squares in this article written for students. Why are they magic?!
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
Different combinations of the weights available allow you to make different totals. Which totals can you make?