Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
To avoid losing think of another very well known game where the
patterns of play are similar.
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
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!
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
Can you find a rule which connects consecutive triangular numbers?
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?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
A game for 2 players
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Can you find a rule which relates triangular numbers to square numbers?
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?
Show that all pentagonal numbers are one third of a triangular number.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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. . . .
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?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
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
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?
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?
Simple additions can lead to intriguing results...
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
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. . . .
Jo made a cube from some smaller cubes, painted some of the faces
of the large cube, and then took it apart again. 45 small cubes had
no paint on them at all. How many small cubes did Jo use?
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?
Place a red counter in the top left corner of a 4x4 array, which is
covered by 14 other smaller counters, leaving a gap in the bottom
right hand corner (HOME). What is the smallest number of moves. . . .
There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being
visible at any one time. Is it possible to reorganise these cubes
so that by dipping the large cube into a pot of paint three times
you. . . .
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
What can you see? What do you notice? What questions can you ask?
What is the shape of wrapping paper that you would need to completely wrap this model?
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?
Have a go at this 3D extension to the Pebbles problem.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Can you describe this route to infinity? Where will the arrows take you next?
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 huge wheel is rolling past your window. What do you see?
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?
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. . . .
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo