I will close this article with three fun examples of paradoxes based loosely on mathematical induction. I qualify my words because in each case induction is not really a pivotal part of the paradox; it appears sometimes merely to grease the wheels of the description, and other times more maliciously, to confuse the reader even more. Moreover, the solutions to these paradoxes do not rely on induction in any particular way, and so they will be omitted entirely! (Or, as mathematicians often prefer to put it, they will be “left as an exercise”.)

First we have the proof that everyone is pretty much bald. It goes by induction: we will prove that for all n, if you have n hairs on your head then you are pretty much bald. The base case is easy: if you have 1 hair on your head, then certainly you’re pretty much bald.

Now suppose inductively that if you have n hairs on your head, then you’re pretty much bald. We need to show that the same is true for someone with n + 1 hairs on their head. But certainly if someone has only 1 hair more on their head than someone else who is pretty much bald, then that first person is also pretty much bald. This completes the induction!

Next we have the proof that every natural number is interesting. Perhaps you already believe this to be so, but a proof certainly wouldn’t hurt! We proceed by induction on n. The base case n = 1 is obvious because of course 1 is a very interesting number. Now suppose that all the numbers from 1 to n are interesting. We need to show that n + 1 is interesting. But consider this: if n + 1 were not interesting then by the inductive hypothesis it would be the very smallest uninteresting number—and that makes it very interesting! This completes the induction.

Finally, we’ll take a quick look at the paradox of the unexpected examination. This one is more serious than the previous two, though how much more is hard to say. The setup is as follows:

Friday afternoon, just before school lets out, a teacher promises his class that they will have a quiz on one of the five days of the coming week. Moreover, he guarantees the students that the quiz will be a surprise in that they won’t be able to predict the night before that it will happen the next day. The class is dismayed until one of the students realizes that something fishy is going on. She reasons:

“The quiz can’t be given on Friday, for sure, because that’s the last possible day, so we would be able to predict it Thursday night. So Friday is out. That makes Thursday the last possible day the quiz can be given. But then the quiz also can’t be given on Thursday, because Wednesday night we would know it was coming the next day! And in the same way we can eliminate Wednesday, Tuesday, and even Monday from the list of possible days for the quiz.”

This argument is good enough to convince the rest of the class, who gleefully go about their business, content in the certainty that there can be no surprise quiz. Tuesday morning comes, however, and the teacher hands out a quiz sheet to each student. There are, of course, ob jections: “You can’t give this quiz! We already figured out that you couldn’t make it a surprise no matter what day you gave it on!”

But the teacher is unperturbed. “You figured that out, did you? Well, here’s the quiz. Aren’t you surprised?” The students reluctantly agreed that they were. But where did their logic go awry?