Problem 996

Solution:

The key is to notice that no loops of size greater than two are possible. For suppose we have $A_1$, $A_2$, ..., $A_n$ with $A_i$ watching $A_{i + 1}$ for $0 < i < n$, and $A_n$ watching $A_1$. Then the distance $A_iA_i$ is greater than the distance $A_iA_{i + 1}$ for $1 < i < n$, and the distance $A_1A_n$ is less than the distance $A_1A_2$. Hence the distance $A_1A_n$ is less than the distance $A_{n - 1}A_n$ and so $A_{n - 1}$ is closer to $A_n$ than $A_1$. Contradiction.

Pick any soldier. Now pick the soldier he is watching, and so on. The total number of soldiers is finite so this process must terminate with some soldier watching his predecessor. If the process terminates after more than two soldiers have been picked, then the penultimate soldier is watched by more than one soldier. But in that case there must be another soldier who is unwatched, because the number of soldiers equals the number of soldiers watching.

If the process terminates after just two soldiers, then we have a pair of soldiers watching each other. Now repeat on the remaining soldiers. Either we find a soldier watched twice (in which case some other soldier must be unwatched) or all the soldiers pair off, except one, since the total number is odd. But that soldier must be unwatched.