# The pop-quiz paradox

This paradox has puzzled me for years. I hoped that Cecil had addressed it, but I was not able to find it in the archive.

Here’s how it goes:

A professor tells his students on the first day of class: “In addition to the scheduled exams, I guarantee there will be one surprise quiz during this semester. By ‘surprise’, I mean that none of you will be able to be certain beforehand that it will occur on a particular day.”

The professor makes two statements:

• A quiz will occur before the end of the semester.
• No one will be able to know for certain that it will occur on any given day.

Let’s say (for the purposes of discussion) that there are 40 days of class in the semester.

Now, the quiz cannot occur on day 40 (the last day of the semester):

1. If the quiz hasn’t occurred by day 39, it would have to occur on day 40.
2. Every student who was paying attention would know it had to be on day 40.
3. This violates one of the professor’s statements, so the quiz cannot occur on day 40.

However, the quiz cannot occur on day 39, either:
4. If it hasn’t occurred by day 38, it must occur on day 39 or 40.
5. It cannot occur on day 40 (as demonstrated in 1-3).
6. So if it hasn’t occurred by day 38, it must occur on day 39.
7. Every student who was paying attention would know it had to be on day 39.
8. This violates one of the professor’s statements, so the quiz cannot occur on day 39.

By continuing this argument, one can demonstrate that the quiz cannot occur on days 38, 37, 36, etc., all the back to day 1. In fact, the argument “proves” the surprise quiz cannot occur at all, when common sense (and personal experience) indicates that it can in fact occur. I have not been able to find the flaw in the argument.

Can anyone help?

This has been discussed several times before:

The term ‘surprise’ can’t be grasped by logic.
So once you have (sort of mathematically) convinced yourself, that the quiz cannot take place on any day, you will be surprised by it whenever it comes. Even if it would be on day 40!

If you want to treat it logically/mathematically you can’t change your opinion in between. So once you decided that there can’t be a quiz, you can’t (let’s say on day 39) change your opinion and say: Oh, now it must be tomorrow.

The paradox ignores the fact that the knowledge that a test has not been given is not available until the end of the lesson.
At the beginning of the lesson there is the option that the test will either be given on that day or on a later day. It is not until the end of that lesson that this knowlege is definite.

actually means by the end of day 38. This means that until the end of day 38 the test can still occur on day 38 or day 39. And so on, and so on.

Actually, I’ve just noticed that you are a relatively new user, so I apologise if I was a little brusque there jhurshman. However, you can use the board’s search facility to see if a topic has been discussed before.

And then, since it’s a logic class, the entire class comes to the same logical conclusion you do, and realize they can’t have a test.

So then, on the 20th day, when the professor gives them a pop quiz, everyone is surprised.

The paradox you described is a pretty well known one. It’s also known as “The Unexpected Hanging”. Here’s one analysis of it:

http://brandt.kurowski.net/projects/lsa/wiki/view.cgi?doc=276

And here’s another (the first page just presents the problem…there’s a link to the solution.

http://www.greylabyrinth.com/Puzzles/puzzle051.htm

Thanks for the pointers. I actually tried searching, but because this paradox doesn’t have a single name, I didn’t find anything. Also, the snail-like speed of the search didn’t encourage multiple attempts.

Here’s a great page for these type things.

http://www.mba.vanderbilt.edu/mike.shor/Courses/game-theory/lecture02.html

I haven’t read any of the links, so this may be covered, but I think the problem occurs here:

in the part where the OP explains why the quiz cannot occur on day 39, it says “It cannot occur on day 40 (as demonstrated in 1-3).” However, one of the premises of 1-3 is that the quiz did not occur on day 39.

Therefore, the proof that the quiz cannot occur on day 39 has as one of its buried premises the statement that the quiz did not occur on day 39.

Taxguy: I don’t think that’s the problem. He might be entirely clear, but I think 1-3 are saying things if it doesn’t occur on day 39.

OP->There’s not a flaw in it, if you are assuming a rational student. The professor may believe he is giving a surpise quiz, but it won’t be a surprise to the student.

Me, I give a variable number of pop quizzes to avoid just such a problem.

How does that avoid the problem? The student may be expecting something that does not happen, but s/he is still expecting it.

a more colourful example along the same vein - the Eye color riddle.

This seems to be similar to the old crim saying to the cop. “I’ll get you when you least expect it.” So the cop says “Then I will always expect it.”

TaxGuy seems to be right - an extra premise is used to show that the quiz cannot take place on day 40, namely that it has not taken place by day 39.

If the premise that it doesn’t occur on day 39 is switched in, it does not work because that does not rule out the possibility that it has not already happened on any other day (including day 40).

Point 5 should read 5. “It cannot occur on day 40 if it has not occurred by day 39.”. Since we are trying to show that it takes place on day 39 the argument falls apart.

I disagree that 5 should read “It cannot occur on day 40 if it has not occurred by day 39.”

If it HAS occurred by day 39, then it cannot occur on day 40 (since there’s only one to give).

If it HAS NOT occurred by day 39, then it still cannot occur on day 40 (as demonstrated in points 1-3).

Therefore, it is unnecessary for point 5 to be “It cannot occur on day 40 if it has not occurred by day 39.”

Furthermore, points 4-8 are not trying to “show that [the quiz] takes place on day 39”. They are purporting to demonstrate that it CANNOT occur on day 39.

The only reason people think this is a paradox is a misunderstanding of the term “surprise”. “Surprise” does not mean that you will never find out; you’re certain to know when the test was after you take it, for instance. But it’s only by assuming that you never find out that you get a paradox.

If the test happens to be on day 40, then it is still a surprise, since the students didn’t know when it would be until the end of day 39.

Chronos: The standard interpretation, which I think is the most reasonable reading of this statement of the puzzle, is that you wouldn’t know the night before that the test would be the next day. Are you sure that’s the resolution?