Parker Glynn-Adey here speaks some about the Pigeon Hole Principle, which is one of those little corners of mathematics whose name alone brings a smile to people’s faces. There are a couple of ways of stating the principle. The version I remember from time immemorial is that if one has N pigeons and a smaller number M of pigeon-holes, then if we’ve put all the pigeons somewhere, there must be at least one pigeon-hole with more than one pigeon.

Glynn-Adey starts from a more general way of describing this situation, and goes through a couple of equivalent versions of the idea, before launching into some of the neat little puzzles that follow directly from this idea. Some of them are nicely surprising and I recommend any of the exercises as a fun pastime.

I admit that when I first learned of the Pigeon-Hole Principle it was in a class that also needed the idea of keeping pigeons on purpose explained to it. We’d have thought more naturally of cubby-holes, but hadn’t ever encountered a cubby.

Below the cut are some pigeon hole related questions I collected together for a Math Circle at the Fields Institute.

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