October 18th, 2005

Splinter: Sensei

Another Puzzle

This is an old one that I think is rather interesting and shows a lot about how we approach logical reasoning:

The Surprise Quiz

One Friday, a professor tells his class of logic students that there will be a surprise quiz next week. It will be a "surprise" because they won't know on which day he will give it.

The students eagerly discuss this among themselves after class and some up with a startling line of reasoning:

The quiz cannot be on Friday. If it were, consider what our situation would be Thursday after class. If we haven't had the quiz by the end of Thursday, then it must be on Friday by a simple process of elimination. But this means that it will no longer be a surprise! We would have deduced when it would be beforehand. Thus, since the professor said it would be a surprise, it cannot be on Friday.

But it likewise cannot be on Thursday. We've already determined that it cannot be on Friday. So if we get to the end of class Wednesday without having the quiz, then, again by the process of elimination, the quiz must be on Thursday. (It's as if, by eliminating Friday, we've moved the end of the week closer.) Since we could determine on Wednesday night that we'd have the quiz on Thursday, though, it is no longer a surprise.

And we can continue this process through the whole week. If we haven't had the quiz Tuesday night, it must be on Wednesday; this eliminates Wednesday as a surprise. It we haven't had the quiz Monday night, it must be on Tuesday; this eliminates Tuesday as a surprise. We're left with only Monday... But it's not a surprise if that's the only possible day! So Monday is eliminated, too.

The students conclude that there can be no quiz. They write this up over the weekend and present it to the professor. Perhaps this is the "real" quiz? The professor compliments them on their work.

The professor then gives his quiz on Tuesday, as he had planned. The students are indeed surprised!

What was the flaw in the students' reasoning?

[Edit: I'm going to post my thoughts tomorrow. There are a number of ways to approach this classic problem. I'll work from the assumption that the professor did not intentionally lie. I also assume (but did not clearly state in the problem) that the professor has selected the quiz day in advance.]