42 Chapter 5

Theorem 34. The answer is no, it is not possible to have a nonempty finite set of peo-

ple pointing at each other in such a way that every person is more often pointed at than

pointing.

Let us give several different proofs.

First proof. Suppose that we have a finite arrangement of people pointing at each other

or themselves. For each person, let us say their pointed-at score is the number of times

someone is pointing at them, and their pointing score is the number of times they are

pointing at someone, including all instances of multiple pointing and self-pointing in both

of these scores. Let A be the sum total of all the pointed-at scores, and let P be the sum

total of all the pointing scores. I claim that P = A. The reason is that every instance of

someone pointing is also simultaneously an instance of someone being pointed at, simply

viewed from the other person’s perspective, at the other end of the finger. Every instance

of pointing adds exactly one to P and also exactly one to A. If every person were more

often pointed at than pointing, however, then it would follow that P < A, since P would be

the sum of a finite sequence of numbers, each of which is smaller than the corresponding

summands giving rise to A. Since P = A, this cannot happen.

Next, we prove the theorem inductively.

Second proof. We prove the theorem by induction on the number of people. That is, no set

of n people can form a counterexample. This statement is true for n = 1 person, since the

person can point only at herself, and if she does so k times, then she will be both pointing

and pointed at k times equally. Suppose now that the statement is true for all groups of size

n, and consider a group of n + 1 people. Suppose that we have an arrangement of the n + 1

people for which everyone is more often pointed at than pointing. Let us call one of those

people “Horatio.” In particular, Horatio is more often pointed at than pointing. Thus, we

may simply remove Horatio from the group of people and direct some of the people who

were pointing at him to point instead at those to whom Horatio had pointed. Since Horatio

was more often pointed at than pointing, there are enough people who had been pointing

at Horatio to cover his pointing commitment. After this rearrangement of the pointing,

anyone left still pointing at where Horatio had been may simply lower his or her finger. In

this way, we arrive at a new configuration, with one fewer person and hence of size n,but

that still satisfies that everyone left is more often pointed at than pointing. This contradicts

the induction assumption that there is no such group of size n, and so we have completed

the induction step. So there can be no such group of people of any finite size.

In the third proof, let us adopt an anthropomorphizing perspective, which enables us

more easily to see the truth of a certain mathematical feature.