Consider a watch face which has identical hands and identical marks
for the hours. It is opposite to a mirror. When is the time as read
direct and in the mirror exactly the same between 6 and 7?
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. . . .
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?
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?
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.
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?
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?
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
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
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?
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?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
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. . . .
A game for 2 players
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. . . .
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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.
To avoid losing think of another very well known game where the
patterns of play are similar.
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
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?
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. . . .
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. . . .
How many different symmetrical shapes can you make by shading triangles or squares?
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
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?
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.
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
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?
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
When dice land edge-up, we usually roll again. But what if we
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Square It game for an adult and child. Can you come up with a way of always winning this game?
Simple additions can lead to intriguing results...
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Can you describe this route to infinity? Where will the arrows take you next?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
What can you see? What do you notice? What questions can you ask?
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which relates triangular numbers to square numbers?
Can you find a rule which connects consecutive triangular numbers?
Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
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.
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
How efficiently can you pack together disks?