Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has exactly one more vertex than it has edges.

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

Explore the transformations and comment on what you find.

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.

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.

Can you think like a computer and work out what this flow diagram does?

What day of the week were you born on? Do you know? Here's a way to find out.

How many different colours of paint would be needed to paint these pictures by numbers?

How many different colours would be needed to colour these different patterns on a torus?

A weekly challenge: these are shorter problems aimed at Post-16 students or enthusiastic younger students.

A weekly challenge concerning the interpretation of an algorithm to determine the day on which you were born.

Our first weekly challenge. We kick off with a challenge concerning inequalities.

Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?

How do different drug-testing regimes affect the risks and payoffs for an athlete who chooses to take drugs?

How can you quickly sort a suit of cards in order from Ace to King?

Look for the common features in these graphs. Which graphs belong together?