First I’ll begin by mentioning that this is an idea completely stolen from a talk by Prof. Imre Leader on games of pursuit, most recently delivered at the ‘Bath’ IMO Squad Nursery camp in Oxford last week.

Suppose we have a ‘lawn’ on which there is somewhere a student. Being in Cambridge, the student is not supposed to be on the lawn, and a group of porters is charged with apprehending the student. The head porter may deploy some number of porters (whose maximum speed is the same as that of the student) to points on the boundary of the lawn and issue them with a strategy to bring the student in (the student must escape in finite time by reaching a point on the boundary at which there is not a waiting porter). Of course, the porters may never step on the grass. The question is, for various lawns, how many porters do you need? The two lawns we will consider are the square shaped lawn in ‘Great Court’, and the circular-shaped law ‘New Court’. As always, I’d advocate the reader sitting down and drawing pictures of the lawns and thinking about (even proving?) how many porters they think they need before reading on.

For the square lawn, it is clear that 4 porters can do the job by just tracking the student’s ordinate parallel to the side they are guarding. A boring check shows that 3 cannot do it.

For the circular lawn, it seems difficult to prove that any number of porters can do it even though it seems that a trillion porters should probably be able to do it. However, the head porter has a cunning plan. There are infinitely many points on the boundary of the lawn, so if he dispatches infinitely many porters, he can just guard every point, so he sends out porters P_1,P_2,...... However, the student realizes that to avoid all the porters, he can avoid each in turn, so it will suffice if the student can get himself to some position really close to the boundary where he will be able to escape P_1 whatever he does next provided he does so quickly, and then in time at most \frac{1}{1000} get himself to a position where he will always escape P_2 provided he escapes by the same deadline he had for P_1, and so on, taking at most 1000^{-(n-1)} time to get to a position from which he will always escape P_n provided he leaves within some sufficient time at least 1000^{-1}+1000^{-2}+... from getting to his position clear of P_1. It’s then clear that, provided his path tends to the boundary, when he eventually escapes none of the porters will be waiting for him. Having had this idea, the problem becomes very simple owing to the arc length of the circle along any distance being greater than the length of the corresponding chord (so if the student is sufficiently near the edge, he can always get well away from any given porter by heading along some straight line). Hence any infinite set of porters labelled with natural numbers fails to catch the student. What has gone wrong in the head porter’s plan? Clearly if we just permenently put a porter on every point of the boundary they are bound to catch the student. The only problem that could possibly have arisen is that he labelled his infinite set of porters with the natural numbers, and the above escape strategy shows that such a set of porters wasn’t actually big enough to implement the obvious strategy and guard every point of the circle. In other words, the head porter has, to his misfortune, discovered that the set of points on the boundary of the circle is uncountable.

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