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### Some Circuits in Graph or Network Theory

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

problem

### Doodles

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

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### Network Trees

Explore some of the different types of network, and prove a result about network trees.

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### Magic Caterpillars

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

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### Tourism

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.

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### Maximum Flow

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

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### Production Equation

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

problem

### flow chart

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

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### Zeller's Birthday

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

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### Painting by Numbers

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

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### Torus patterns

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

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### Happy birthDay

Can you interpret this algorithm to determine the day on which you were born?

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### The Olympic Torch Tour

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

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### Drug testing

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