If a teacher says he is going to give you a quiz the following week but the day will be a surprise. Can it really be a surprise?
No, the day cannot be a surprise. First, imagine that if you make it through Thursday and there has been no quiz, then it would have to be given on Friday. So in that situation, you would not be surprised when he pulls out the quiz on Friday. Therefore, a surprise quiz can't happen on Friday.
Since it can't be a surprise on Friday, the last day it could possibly be a surprise would be Thursday. At the end of Wednesday, you know the quiz has to be on Thursday to be a surprise, so when the quiz does come on Thursday, it was the only possibility so it won't be a surprise. So it can't be on Thursday either.
Following that same logic, it could not be on Wednesday, or Tuesday, or Monday either. So apparently, the teacher cannot give a surprise quiz. Yay for you!
Since you realize he can't give a surprise quiz, you decide not to study for it. Then on the day he actually does give the quiz, boy will you be surprised!
This is called "the unexpected hanging paradox" and is explained here.