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?
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?
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. . . .
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
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 bicycle passes along a path and leaves some tracks. Is it
possible to say which track was made by the front wheel and which
by the back wheel?
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
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?
A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
A spider is sitting in the middle of one of the smallest walls in a
room and a fly is resting beside the window. What is the shortest
distance the spider would have to crawl to catch the fly?
Draw all the possible distinct triangles on a 4 x 4 dotty grid.
Convince me that you have all possible triangles.
A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?
How efficiently can you pack together disks?
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?
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
A cube is made from smaller cubes, 5 by 5 by 5, then some of those
cubes are removed. Can you make the specified shapes, and what is
the most and least number of cubes required ?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
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. . . .
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
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?
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?
A right-angled isosceles triangle is rotated about the centre point
of a square. What can you say about the area of the part of the
square covered by the triangle as it rotates?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
It is known that the area of the largest equilateral triangular
section of a cube is 140sq cm. What is the side length of the cube?
The distances between the centres of two adjacent faces of. . . .
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.
See if you can anticipate successive 'generations' of the two
animals shown here.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
What can you see? What do you notice? What questions can you ask?
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
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Can you maximise the area available to a grazing goat?
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 happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?
This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the. . . .
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .
Two intersecting circles have a common chord AB. The point C moves
on the circumference of the circle C1. The straight lines CA and CB
meet the circle C2 at E and F respectively. As the point C. . . .
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?
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
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. . . .
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. . . .
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
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?