More pointed at than pointing

A selection from Proof and the Art of Mathematics

Gather yourself and a few friends into a circle and point at each other in some arrangement of pointing. Let each person point at one or more of the others, or at themselves, or at nobody, as they like. Use both hands, or different fingers, or your feet if you want to point at several people, and let us say that it is allowed to point more than once at a given person, or at several people, or at yourself—go to town! Perhaps some people are pointing quite a lot and others are pointing much less, and similarly with being pointed at.

Now, I have a question about whether you might be able to achieve a certain feature in your pattern of pointing at each other.

Question: Can you arrange it so that every person is altogether more often pointed at than pointing?

In other words, could we all be pointed at strictly more times than we point at others? Ponder the problem on your own, before reading further.

Interlude

This is a brief excerpt from chapter 5 of Proof and the Art of Mathematics, an introduction to the art and craft of proof-writing for aspiring mathematicians who want to learn how to write proofs. The book is filled with compelling mathematical statements having interesting elementary proofs.

Proof and the Art of Mathematics has been awarded the 2024 Daniel Solow Author’s Award by the Mathematical Association of America, awarded for outstanding and impactful contributions to mathematics education.

Theorem. The answer is no, it is not possible to have a nonempty finite set of people 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. □

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. □

Third proof: Suppose we are part of a finite group of people pointing at each other, and everyone is more often pointed at than pointing. Let us instruct everyone to pay one dollar each to the people to whom they point, for each instance of pointing; and let us assume that we all have enough cash on hand to do this. The curious thing to notice is that, after the payments, because everyone is more often pointed at than pointing, it follows that every person will take in more money than they paid out. We made money! And we could do it again and make more money, and again and again, as many times as we desire. We could make millions of dollars simply by exchanging it like this. Since this is clearly impossible, as the total dollar holdings of the group does not change as money is exchanged within it, there can be no such pointing arrangement. □

I find the third proof very clear, though I recognize that it is essentially similar to the first proof, if one simply thinks of the pointing-at and pointing scores as measured in dollars. Perhaps the reason it is so clear is that it replaces the abstract quantity-preserving argument of the first proof with something much easier to grasp, namely, the fundamental fact that we cannot get more money as a group simply by exchanging money within our group. Such anthropomorphizing arguments or metaphors can often be surprisingly effective in simplifying a mathematical idea. We leverage our innate human experience in order to understand more easily what would otherwise be a complex mathematical matter. Our human experience with the difficulty of getting money makes the final conclusion of the third argument obvious.

Selections from the end of the chapter:

Mathematical Habits

Use metaphor. Express your mathematical issues metaphorically in terms of a familiar human experience, if doing so makes them easier to understand. Find evocative terminology that represents your mathematical quantities or relationships in familiar terms, if doing so supports the mathematical analysis.

Exercises

1. Suppose that a finite group of people has some pattern of pointing at each other, with each person pointing at some or all or none of the others or themselves. Prove that if there is a person who is more often pointed at than pointing, then there is another person who is less often pointed at than pointing.

2. Suppose that you could control who follows whom on Twitter. Could you arrange it so that every person has more followers than people they follow? For example, some extremely famous people currently have many millions of followers, and one might hope to reassign most of those followers in such a way that everyone will be more followed than following.

3. Show that if there are infinitely many people, then it could be possible for every person to be more pointed at than pointing. Indeed, can you arrange infinitely many people, such that each person points at only one person but is pointed at by infinitely many people? How does this situation interact with the money-making third proof of theorem?

Credits. The pointing-at painting is David and Charles Colyear by Sir Godfrey Kneller, in the public domain via Wikimedia Commons.