The dot without arrows

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.

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