This argument is known as a backwards induction argument. Despite this argument a surprise exam is obviously a possibility. We must assume the pupils are completely rational and not suffering any defects in memory, and they know they are rational and have good memories. There is no puzzle otherwise , the forgetful and stupid can obviously be surprised.
But does the backwards induction argument get started. They know on Thursday evening that they will think ' so either there will be an exam which I expect or they'll be no exam. But in that case I can no longer be sure there'll be an exam, since the teacher's announcement cannot be fulfilled. So it could be a surprise after all, but then I'll expect it so it won't be a surprise '
This could go around and around indefinitely. In such an unstable position the pupil cannot be sure there will be an exam, so if it does happen it'll be unexpected. In consequence the argument does not get started. There can be a surprise exam even on Friday.
Suppose however the pupils can be certain there will be an exam, it is an exam that takes place every year, and it is unthinkable it will be cancelled. Suppose also they have good reason to trust the teacher and accept it will be a surprise, although an expected exam is not as unthinkable as no exam. In this case the exam cannot be on Friday because if there is no exam on Thursday they'll expect it on Friday. So the argument does get started this time, but doesn't get very far. On Wednesday evening they will think, since Friday is ruled out , there is only one possible day for the exam, tomorrow, but then it will be expected. If the teacher's word is not going to be fulfilled and the exam will be no surprise, there are still two days on which it may be held, and we have no way of choosing between them. But then it can be a surprise tomorrow but not Friday, but then we shall expect it tomorrow, so it won't be a surprise
This reasoning can also go on indefinitely, in such an unstable position they cannot be sure the exam will occur on Thursday so they can be surprised if it's given then, after that it is too late. This paradox was traced back to wartime Sweden and a radio broadcast in which a surprise civil service defence exercise was announced for the following week. From book paradoxes a to z by Michael Clark
