What do you think about the following question?
"There are 99 spies, one looking to the nearest other. Is there one spy who is not looked at by any other."
I think the answer is yes, there is one such spy. Going recursively, with 1 spy, nothing interesting happens. With 2 spies, they look at each other. With 3 spies, one can select the one furthest away from all other ones, he is the one not looked at by any other (unless the spies are at the corners of a equilateral triangle). Going on like that the statement is true for 99 spies.
Is the above line of reasoning correct?
Let there be $n>3$ spies in total.
If we were to assume that distances can not be repeated, then among all of the spies and their positions, $x_1,x_2,\dots$ then among the distances between them $d(x_1,x_2),d(x_1,x_3),\dots$ there will be a unique minimum found among them (unique as guaranteed by the assumption that distances can not be repeated).
The two spies corresponding to said minimum distance will be looking at each other. Without loss of generality suppose the two spies with minimum distance apart happened to be spies $x_n,x_{n-1}$. When considering all other spies apart from these two, if any were looking at either of those two spies there will be strictly more outgoing "looks" than incoming among spies $x_1,\dots,x_{n-2}$ and so we would necessarily have at least at least one not looked at.
So, otherwise, suppose that none of $x_1,\dots,x_{n-2}$ are looking at $x_{n-1}$ or $x_n$. This effectively puts us back in the original scenario and we can continue by induction reducing the number of spies by two repeatedly until finally arriving either at the case of $n=2$ or $n=3$. In the case of $n=2$ it is indeed possible that they look at each other and we end with no spies not being looked at, implying that if we start with an even number of spies it is possible that all are looked at by somebody (and occurs precisely when all spies occur in pairs).
Otherwise, in the case of $n=3$ we will again have two of these three who have minimum distance to one another looking at one another. The remaining third spy however will not be looked at by anyone (again, emphasizing that we assumed distances between spies to be unique).
This proves that whenever $n$ is odd (and when distances between spies is unique) we necessarily have at least one spy not looked at.
In the event that distances are not necessarily unique, and the decision of who to look at is arbitrary in the event of ties (or that you look at all nearest neighbors), with spies standing in a regular $n$-gon you can have all spies looked at in the case of odd $n$, or a regular $(n-1)$-gon in the case of even $n$ and one isolated spy to have a spy not being looked at.