You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
A cheap and simple toy with lots of mathematics. Can you interpret
the images that are produced? Can you predict the pattern that will
be produced using different wheels?
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
Can you maximise the area available to a grazing goat?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
I found these clocks in the Arts Centre at the University of
Warwick intriguing - do they really need four clocks and what times
would be ambiguous with only two or three of them?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
See if you can anticipate successive 'generations' of the two
animals shown here.
Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?
What can you see? What do you notice? What questions can you ask?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
What is the shape of wrapping paper that you would need to completely wrap this model?
Take a line segment of length 1. Remove the middle third. Remove
the middle thirds of what you have left. Repeat infinitely many
times, and you have the Cantor Set. Can you picture it?
A rectangular field has two posts with a ring on top of each post.
There are two quarrelsome goats and plenty of ropes which you can
tie to their collars. How can you secure them so they can't. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
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?
Each of the nets of nine solid shapes has been cut into two pieces.
Can you see which pieces go together?
These are pictures of the sea defences at New Brighton. Can you
work out what a basic shape might be in both images of the sea wall
and work out a way they might fit together?
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
Draw a pentagon with all the diagonals. This is called a pentagram.
How many diagonals are there? How many diagonals are there in a
hexagram, heptagram, ... Does any pattern occur when looking at. . . .
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
Which of the following cubes can be made from these nets?
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start.
How many Hamiltonian circuits can you find in these graphs?