A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
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?
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?
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
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?
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?
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?
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.
This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!
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. . . .
Imagine a stack of numbered cards with one on top. Discard the top,
put the next card to the bottom and repeat continuously. Can you
predict the last card?
Glarsynost lives on a planet whose shape is that of a perfect
regular dodecahedron. Can you describe the shortest journey she can
make to ensure that she will see every part of the planet?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
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 ?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
An irregular tetrahedron has two opposite sides the same length a
and the line joining their midpoints is perpendicular to these two
edges and is of length b. What is the volume of the tetrahedron?
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
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?
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.
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. . . .
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?
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.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
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?
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 game for 2 players
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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 article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
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. . . .
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
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. . . .
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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
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 circular plate rolls in contact with the sides of a rectangular
tray. How much of its circumference comes into contact with the
sides of the tray when it rolls around one circuit?
What can you see? What do you notice? What questions can you ask?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?