Nyet's Brain Echo, Vol. 1

A classic, courtesy of a paper I was reading by Timothy Chow:

A teacher announces in class that an examination will be held on some day during the following week, and moreover that the examination will be a surprise. The students argue that a surprise exam cannot occur. For suppose the exam were on the last day of the week. Then on the previous night, the students would be able to predict that the exam would occur on the following day, and the exam would not be a surprise. So it is impossible for a surprise exam to occur on the last day. But then a surprise exam cannot occur on the penultimate day, either, for in that case the students, knowing that the last day is an impossible day for a surprise exam, would be able to predict on the night before the exam that the exam would occur on the following day. Similarly, the students argue that a surprise exam cannot occur on any other day of the week either. Confident in this conclusion, they are of course totally surprised when the exam occurs (on Wednesday, say). The announcement is vindicated after all. Where did the students' reasoning go wrong?

Ponder this one. This puzzle and another (to be revealed soon) have been eating away at my brain the past couple of days, not because of the ooh-ahh nature, but the "how do you actually formulate the argument" nature. Clearly something is going on with the word "surprise" and the impact of knowledge. More later.

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UPDATE
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It seems clear, especially with the help of Margie's link, that this is a problem that even the headiest of modal logicians and epistemological theorists cannot solve. Rather than bore my already bored audience with the details, suffice it: our seemingly simple statements like those above are worm-cans of confusion. Consider this next time someone unverifiably tells you that they *know* something.