The Pigeonhole Principle

A frigid Saturday morning, linking around the internet, catching up on stored articles set aside during a busy week. Hit on an article on a couple scientists bridging biological neurology and artificial intelligence, which leads to questions about higher mathematics and logic which in turn lead to this delightful principle of which I had not been formally aware but which is seductively obvious, the Pigeonhole Principle. From Wikipedia:

In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.[1] This theorem is exemplified in real-life by truisms like "there must be at least two left gloves or two right gloves in a group of three gloves". It is an example of a counting argument, and despite seeming intuitive it can be used to demonstrate possibly unexpected results; for example, that two people in London have the same number of hairs on their heads (see below).

I find it fascinating how there are principles lurking behind knowledge, ideas which were on the cognitive horizon which, once pointed out, are obvious. This is one of those.

Some thirty or forty years ago I learned that if there are 23 people in a room, there is a greater than 50% probability that two of them have the same birthday. When I first heard this, I sort of lazily muscled through the abstract ideas behind the forecast in order to verify to my own satisfaction that the claim was likely true. But once satisfied, I have never really thought about it again. I certainly never sat down and worked out the specifics.

Now, from an article about modelling the brain, by many linking ideas, I discover that it is the Pigeonhole principle which underpins the forecast that there is a 50% chance that, in a room of 23 people, two of them will have the same birthday. Again from Wikipedia:

The birthday problem asks, for a set of n randomly chosen people, what is the probability that some pair of them will have the same birthday? By the pigeonhole principle, if there are 367 people in the room, we know that there is at least one pair who share the same birthday, as there are only 366 possible birthdays to choose from (including February 29, if present). The birthday "paradox" refers to the result that even if the group is as small as 23 individuals, there will still be a pair of people with the same birthday with a 50% probability. While at first glance this may seem surprising, it intuitively makes sense when considering that a comparison will actually be made between every possible pair of people rather than fixing one individual and comparing them solely to the rest of the group.

Wonderful.