Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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?

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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?

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?

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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 ?

What can you see? What do you notice? What questions can you ask?

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?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

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. . . .

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

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. . . .

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

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?

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.

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?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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?

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?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?

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.

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?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

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?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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?

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.

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?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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?

Can you find a way of representing these arrangements of balls?

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?

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. . . .