The Surprise Quiz Paradox, Solved

2019.03.13   prev     next

SO the story goes, a logic professor tells her five-days-a-week class, “This coming week, there will be a surprise test. I promise you that it will be a sur­prise.”

Meeting at a pub after class, the students ponder what the professor meant. One clever student announces that they don’t need to study at all. “She can’t give the test on Friday, because then it wouldn’t be a surprise. So Thursday is the last day she can give it. But then she can’t give it on Thursday, because it wouldn’t be a surprise. So Wednesday is the last day she can give it. And so on, all the way back to Monday. So there’s no day she can give the test! That must be what she wanted us to figure out; we don’t have to study.”

The students rejoice, drink hearty, and go to class the next week confident that there will be no test. Monday there’s no test, Tuesday there’s no test — but on Wednesday, the professor pulls out the test. “Surprise!” she says. And indeed, the students are very surprised.

What was wrong with the clever student’s logic?

• • •

Part of the solution has to do with hammering out ambiguities in the definition of the problem, but first, the straightforward answer:

The flaw in the student’s logic is that he is conflating (equivocating between)

things the professor can choose to do

and

the professor keeping her promise.

These two categories are not the same, and there is no way to exactly define the problem such that those two categories are the same.

First, let’s explore the premise that the professor can choose to not give the test at all. (That, after all, is implied by the student’s conclusion.)

If she doesn’t give the test at all, she’s breaking her promise. But she can choose to do that. In that case, the student’s logic fails on the very first step, because the professor can give the test on Friday, and it will be a surprise. Here, we precisely define surprise as the students not knowing with extremely high (99% certitude) that the test will occur that day. Since, as far as the students know, she might not give the test at all, then a test on Friday would count as a surprise.

That solves the whole puzzle right there. But if you’d like a deeper analysis, consider this next prem­ise:

Suppose that the students all have computers at their desks, which can’t be closed or turned off, and suppose the college’s class administration software is programmed to force this week’s exam to occur on Friday if it hasn’t been given already on an earlier day that week. The professor can choose to give it on an earlier day, but if she hasn’t done that when class convenes Friday, then the quiz will automatically begin; the professor can’t stop it or intervene in any way. The grade will count, whether the professor wants it to or not. And suppose that this programming is open-source; it’s common knowledge among professors and students alike.

In that case, the clever student’s logic survives its first step, “If she gives it on Friday, it won’t be a surprise.” But it then fails on the very next step, the same way it failed in our previous premise: The test will be a surprise if given on Thursday, because the students don’t know that it won’t be given on Friday (even though it wouldn’t be a surprise then).

(Also, it should be noted that if it is established by computer-programmed policy that the test absolutely will occur this week, then any conclusion that “we don’t need to study” is obviously incorrect.)

You can make things even more complicated by supposing a computer program that doesn’t permit the professor to give the test on Friday, and will therefore force the test to happen Thursday if not already. But that just takes the student’s logic one step further. If you keep backing up the code until the student’s logic survives all the way, then you have created a computer program that doesn’t permit the professor to give the test at all. In that case, the student’s conclusion is indeed correct, and the test does not occur.

One final note. The cute story at the end — about the professor giving the test on Wednesday and the students being very surprised — is exactly that: a cute story. It makes the “paradox” feel even more bafflingly paradoxical, but it’s purely anecdotal: it proves nothing at all.

And there you have it. Everything you wanted to know (or perhaps didn’t) about the Surprise Quiz Paradox.

See also:
Why Quintic (and Higher) Polynomials Have No Algebraic Solution
&
The Digits of Pi Cannot Be Sequentially Generated By A Computer Program

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