This is one of my all-time favorite paradoxes:
On Friday, a teacher announces to his class that there will be a pop quiz one day during the following week. In order to uphold the integrity of the quiz, it must satisfy the following two conditions:
- The quiz will be handed out at the beginning of class one day the following week (Monday through Friday).
- The students will not be able to (logically) deduce which day the quiz will be held before they are actually given the quiz.
After making this announcement, he notices a lot of murmuring around the class, and after a minute a few students start sniggering. He asks the students what they find so amusing. One of the cleverer students in the class raises her hand and explains:
“You cannot possibly give us a pop quiz next week,” she says.
“Why not?” the teacher asks.
“Well, if we won’t be able to deduce which day the quiz is before we are given the quiz, it can’t be on Friday. For if after class on Thursday we still haven’t gotten the quiz, we will know that the quiz must be the following day. This contradicts the second requirement for a pop quiz.”
“That makes sense,” he agrees.
“So the quiz can only be given one of the days from Monday through Thursday. By the same argument as before, the quiz can’t be on Thursday: Since we’ve shown it can’t be held on Friday, we will know by Wednesday if the exam is to be held on Thursday. By the same token, the quiz can’t be held on Wednesday or Tuesday. The only possible day you could give the quiz is Monday. But I know now that the pop quiz must be on Monday, which again contradicts the second condition for pop quizzes. So you can’t give us a pop quiz next week!”
The teacher tells the class that he doesn’t see any flaw in the student’s argument, so it appears that he can’t satisfy the requirements for a pop quiz. The students seem pleased to have a weekend free of study having deduced that a pop quiz is an impossibility. Imagine the students’ surprise when the teacher hands out the pop quiz on Tuesday at the beginning of class, thus fulfilling the previous week’s proclamation!
I first encountered this paradox in a collection of Martin Gardner‘s writings entitled The Colossal Book of Mathematics, where it is referred to as “The unexpected hanging.” A more detailed account of this paradox can be found in this article by Timothy Chow. The subtlety of the paradox is suggested by the sheer number of references cited in Chow’s essay.