So today is the last day of classes before finals. We spent today talking about the Pigeonhole Principle in Discrete Math. The Pigeonhole Principle is one of my favorite math-related things. I’ve done a blog on the actual Principle before, but while we were talking about it today I think I realized why the conclusions you can reach from the Principle seem so counter-intuitive to people (or at least why it was so to me when I first learned of it).

Let’s take a fairly simple example to demonstrate.

Suppose I have a group of 27 individuals. According to the Pigeonhole Principle, I can state that at least two of these people will have names that start with the same letter of the alphabet. I won’t go into why this is so (you can read my version of the explanation here on my previous PP post, linked above), but even if you’re pretty familiar with it, this still seems a little counter-intuitive, doesn’t it? You think, “wait, how can that possibly be a valid conclusion? There’s no way we can guarantee that!”

Where does this aversion to this conclusion come from?

Well, originally for me, I realized it stemmed from how I actually interpreted the conclusion itself. I always automatically interpreted the conclusion as claiming, in the case above: “there will be at least one name that starts with every letter of the alphabet in my group of 27 people.”

Which, of course, is not what the conclusion says at all. There is no claim made about the dispersion of the number of names per letter other than the fact that at least one letter will be the first letter in two names. I could have the case where I have a single name beginning with each letter A-Y, and then two names that begin with the letter Z. That still fits with the conclusion. However, I could also have NO names that start with a letter A-Y and have all 27 start with the letter Z. That’s valid, too. All the conclusion tells us is that at least one letter will begin two names. It doesn’t say that all the letters have to start a name (that is, it doesn’t say that all “pigeonholes” actually have to be utilized).

Now that I’ve typed that out, it seems like a really stupid reason for having trouble understanding the Principle, so it’s probably just me who has this issue. But anyway.

Isn’t this Principle COOL either way?!