1. Let O be the centre of the circle, L the lion and M the Christian. What happens if L keeps on the radius OM and approaches M at top speed?

3. What about a random rotation?

For the second part, use a suitable approximation.

4. Can the line be partitioned into countably many closed sets?

5. Try local changes.

7. Imagine that you can use negative amounts of fuel as well (i.e., can get a loan for your future intake), and every time you get to a town, you get a new lot of fuel. What happens if you go round and round the circuit?

10. Ask the same question for a ‘suitable’ countable set.

11. Which sums seem to be most likely to come up, and which ones seem least likely?

14. Show that the following greedy algorithm constructs a sufficiently large independent set. Pick a vertex x1 of minimal degree in G1 = G, and let G2 be the graph obtained from G1 by deleting x1 and its neighbours. Pick a vertex x2 of minimal degree in G2, and let G3 be obtained from G2 by removing x2 and its neighbours. Proceed in this way, stopping with Gℓ and xℓ, when Gℓ is a complete graph.