For internal use NOT for publication

Suppose that n is even, say 2m. It could happen that there are m pairs of people with each pair close together and the pairs relatively far from other pairs. Then in each pair, they will shoot at each other and everyone will get wet. Thus if there is always, whatever the configuration, someone who stays dry, then n must be odd.

If we allow equal distances between people then it could happen for any n that everyone gets wet. For example they could be equally spaced around a circle and each shoot their neighbour going round in a clockwise direction.

Let us now consider n is odd, and there are n people such that the distance between any two of them is different from the distance between every other pair. The proof that someone stays dry for n=3, 5, 7, 9, ... is by induction.

The case for n=3 is easy. Say the people are p₁, p₂ and p₃ where the distance between p₁ and p₂ is less than the distance of either of these two from p₃ then p₁ and p₂ will shoot at each other and p₃ will stay dry.

Now suppose that the conclusion is true for 3, 5, 7, 9, ..., m (with m odd) and consider the case for m+2 people. There is a pair that are closer together than any other pair and closer together than either of them is to any other individual so they must shoot at each other. If we remove them from the field it makes no difference to what happens to the rest. Amongst the remaining m people the induction hypothesis tells us that one must stay dry so amongst the m+2 people one must stay dry. So if the result is true for m it is also true for m+2 and as we have shown it to be true for 3 it must be true for all odd numbers.