There are several different versions of this particular paradox – like Holli’s Paradox, and The Unexpected Hanging Paradox. Before I present the surprise examination version, it behooves me to mention that no correct (or final) solution to this paradox has been established yet.
Here’s the paradox:
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?