The dot without arrows

Dots and connecting lines

When I was a teenager I found in some now-forgotten magazine a math problem that intrigued me at the time. The magazine very briefly described the scenario something like this: Imagine a sheet of paper that has a number of dots scattered across it in a random layout. If there are n dots on the sheet, how many dots are not the closest neighbor of their closest neighboring dot?

The problem must have been presented in an earlier edition of the magazine, because the article I read briefly described the problem and then walked through ideas from readers on how best to answer it. The largest number of responding readers reasoned that the answer was 0 if n is even, and 1 if n is odd. That made a lot of sense to me as well, because if Dot A’s closest neighbor is Dot B, then wouldn’t Dot B’s closest neighbor be Dot A as well? It turns out, however, that in math—as in life—not everything is so simple.

As a prelude to the correct answer, the magazine explained that a good way to visualize the problem was to draw connecting arrows from each dot to the dot closest to it. When the dots are given a random placement, yes, there will end up being a fair number of pairs of dots with arrows going both ways, but there will also be a smattering of clusters of dots where one of them has more than one incoming arrow, which of course also means there will be some lone dots with only outgoing arrows as well as some dots being pointed to by dots that the first dot is not pointing back to. I probably made that sound more complicated than it really is. The important part is that although each dot will have one arrow going out from it to another dot, any dot without a reciprocal arrow returning to it—from the same dot its arrow is pointing to—is not the closest neighbor of its closest neighbor and is to be counted in the final answer.

Now, I’m not going to bore you with the detailed math equation that solves the stated problem because a) it’s pretty advanced math, and I certainly didn’t fully understand it at the time; b) this many years later, I’m not sure I would remember it exactly even if I had understood it; c) this blog is about life, not math; and d) even if I were misdirected enough to think I should give you the answer, you’re probably even less inclined toward serious math than I am, and you would stop reading this post if I tried to walk through the solution. I present the problem to you now only to describe how I felt when I read it.

Perhaps I tend to anthropomorphize too readily, but in my teenage mind, what struck me most was this idea of a dot sitting there on the page, sending out its arrow, waiting, hoping, excitedly, tentatively, for an arrow back … only to get nothing. How would such a dot feel? It wasn’t the dot’s fault it was placed where it was on the page. Other than sheer dumb luck, why was it alone? What was a dot to do when it sent its arrow in one direction, but got an arrow pointing back to it from an entirely different direction? And what did the dots with many incoming arrows do to merit their popularity?

Maybe I thought of it this way because, as a teenager, I was one of the lone dots. The three years of junior high school, which is when I would have read this magazine article, were the hardest of my school years. I didn’t have any friends going into junior high, and I didn’t make any friends while I was there. It’s not that I didn’t want friends. There was still an arrow pointing out from my dot. But I didn’t have any arrows pointing back.

Taking a broader view of my life, I recognize that I have pretty much always been a loner. As a child, to some degree. As a teen, certainly. Even as a married man I soon learned that Ex-Wife didn’t have her arrow pointing to me. And Girlfriend? Although my arrow was pulled toward her more strongly than I’d ever felt for another human being, it turned out that she was one of those dots with many incoming arrows, and she chose to reciprocate in a different direction.

I’ve spent a lot of time, over the years, wondering at this strange phenomenon. In the case of the dots drawn randomly on a sheet of paper, any individual dot could take no special pride in its popularity or blame for its loneliness. It was, after all, just a question of luck.

Life, though, isn’t a static piece of paper, and we are not dots placed randomly. Instead, life is a frenetic swirl of dots constantly in motion, arrows forming and dissolving and reforming all the time. Of course, as I have also learned in the years since I was a teenager, life isn’t as binary as I was taught to believe. I grew up with the idea that happiness consisted in one dot finding a compatible dot and then drawing and maintaining strong connecting lines between the two. But I now know that there exist dots that are not satisfied with just a single arrow.

In addition, in life there are a million variables that go into whether anyone else will like us or want to spend time with us. Surely, the logic must go, there are enough of those variables over which I have control that I was largely responsible for my past conditions, that I must assume the burden of responsibility for my current state, and that I could, with enough skill and foresight, change my future from being a dot with only an outgoing arrow, to one with a matching incoming connection.


Or maybe not. Maybe there’s just something about us over which we don’t really have any control, something as unchangeable for us as the location in which a dot has been randomly yet irrevocably drawn on a sheet of paper. Each dot on the page has an x and a y that it can’t change. Maybe we have a through z—or aaa through zzz—that we can’t.

Perhaps it’s a bit like that seemingly universal experience in grade school where all the children are lined up on the playground and two team captains are picking from among them, one at a time, for kickball. There’s always that one kid who gets picked last. And no matter how many times this scenario plays out, sport after sport, year after year, it’s always the same kid nobody wants.

It seems to me that all the school children inherently understand why. Nobody has to discuss it. Nobody has to explicitly state what qualities make that kid undesirable. No matter which two kids are chosen as team captains, it’s always that same kid who gets picked last. Everyone just gets it.

Everyone, that is, except for that kid. He can’t figure it out. He doesn’t know what he has to do differently to be chosen.

The truth of it is, though, it doesn’t matter what he does differently. He still won’t be chosen. He doesn’t have what it takes. Sports is just not his thing.

You wish someone would explain this to him. You wish someone would say to him: “Stop caring about this. There are more areas of life than sports.” Because you know he’ll never be happy standing in that line over and over again.

I know, because I’ve been that kid.

I know, because I still am that kid.

The kid nobody picks.

I’m included in the number that answers the math problem. I’m a dot with an arrow pointing to a dot whose arrow is pointing somewhere else. Being a dot without a reciprocal arrow is more common than you might think. You thought I wouldn’t tell you the answer, but I’ve done the math: F(n) = 0.3785n. In other words, roughly 38% of the dots are unpaired. It gets worse, though. That’s the answer for a dot in just two dimensions: x and y. As you increase the number of dimensions, say, for example, to the potentially millions of human attributes, the answer becomes F(n) = 0.5n. The same odds as the flip of a coin.

I think it’s time to accept the reality of the situation. Since nobody else seems to want to, it’s time for me to tell myself: “Stop caring about this. There are more areas of life than relationships.” It’s time to stop waiting for an incoming arrow that is never going to materialize. And time, too, to stop sending out an arrow to a dot that is pointing its own arrow somewhere else.

Pretty much all my life I’ve been the dot with no incoming arrow. Now I need to be the dot without any arrows at all.

I just need to figure out how.

That’s my truth. What’s yours?