This was written in 2009. I explain in this essay why I think the surprise exam paradox is just a subtle version of the paradox of the heap. This was not easy to figure out at all because this is one of the most confusing paradoxes on the planet.
The surprise exam is a cheeky and confusing paradox which dates to the 1940s. Easy to grasp but maddening to resolve, it is one of the most tantalizing paradoxes around.
A teacher tells her class that she will hold an exam some time next week, between Monday and Friday, on a day the class will be unable to anticipate beforehand. In other words, the class will know the exact day of the exam (e.g., Wednesday) only when that day arrives.
Upon hearing this, a clever student thinks for a moment, then argues that no such surprise exam can be held, i.e., that the teacher cannot keep her word!
He starts by pointing out that the teacher obviously can’t hold the exam on Friday, the last day of the week. For if she did, the class would know by Thursday’s end that it was going to occur on Friday, since no exam would so far have occurred and only one day would be left for the exam to be held. But she said that they would not know the day of the exam beforehand. So if she wants to keep her word, she can’t hold the exam on Friday.
But now it follows that she can’t hold the exam on Thursday either. For if she did, the class would know by Wednesday’s end that it was going to occur on Thursday, since no exam would so far have occurred and only two days would be left, with one of them (Friday) already known to be ruled out! Again, they’d know the exam day beforehand. So if she wants to keep her word, she can’t hold the exam on Thursday either.
By the same reasoning, she can’t hold the exam on Wednesday, Tuesday or even Monday, without breaking her word. Apparently, the surprise exam can’t be held at all!
Something is obviously wrong with the student’s reasoning, but what? A precise explanation is not easy to achieve.