Surprise exam paradox?

I just remembered about a problem/paradox I read years ago in the fun section of the newspaper, which has had me wondering often times. The problem is as follows:

A maths teacher says to the class that during the year he'll give a surprise exam, so the students need be prepared the entire year. One student starts thinking though:

1. The teacher can't wait until the last day of school, because then the exam won't be unexpected. So it can't be the last day.
2. Since we've removed the last day from the list of possible days, the same logic applies to the day before the last day.
3. By applying 1) and 2) we remove all the days from the list of possible days.
4. So, it turns out that the teacher can't give a surprise exam at all.

Following this logic, our student doesn't prepare for this test and is promptly flunked when the teacher does give it somewhere during the middle of the year (but that's my own creative addition to the problem).

This problem reminds me about the prisoner's dilemma for finite number of turns - you have to betray at the last turn because tit-for-tat retaliation is no longer relevant (no next turn), but then that means that you have to betray at the turn before that, and so on, until you reach the conclusion that you can't cooperate at all.

So is the student's reasoning correct or not? Mathematically it looks like it should be, but that would imply that surprise exams are not possible (and they are).

Solution 1:

There is a model of knowledge, essentially due to Robert Aumann, in which knowledge is represented by a partition $\Pi$ of a set of states of the world $\Omega$. If the true state of the world is $\omega$, the agent with partition $\Pi$ only knows that some state in the cell $\pi(\omega)$ (the value of the projection at $\omega$) obtained. An event is simply a subset of $\Omega$. We say that an agent knows that the event $E$ obtains at $\omega$ if $\pi(\omega)\subseteq E$. Now let the state space be $\Omega=\{1,2,\ldots,T\}$, where we interpret $t$ as "there is an exam at $t$". Now there is no partition $\Pi$ such that the following holds:

1. The student doesn't know exactly at which date the exam is at any state.
2. If there was no exam at $\{1,\ldots,t-1\}$, then the student knows this at $t$.

Proof: Let $t$ be an element in $\Omega$ such that $\pi(t)$ is not a singleton. Such an element must exist by 1. Let $t'$ be the largest element in $\pi(t)$. By assumption $t'>t$ and so by 2., $\{1,\ldots,t'-1\}$ is a union of cells in $\Pi$ that contains $t$. Since $\Pi$ is a partition, $\pi(t)\subseteq\{1,\ldots,t'-1\}$, contradicting $t'\in\pi(t)$.

So at least using the model of knowledge used above, the surprise exam paradox cannot be formulated coherently.

Solution 2:

A very nice discussion of the unexpected hanging paradox can be found in chapter 43 of Martin Gardner's The Colossal Book of Mathematics (New York: W. W. Norton & Company, 2001). Numerous references are included.

Gardner, citing O'Beirne, states that "the key to resolving the paradox lies in recognizing that a statement about a future event can be known to be a true prediction by one person but not known to be true by another until after the event."

The teacher giving the surprise exam "knows that his prediction is sound. But the prediction cannot be used to support a chain of arguments that results eventually in discrediting the prediction itself. It is this roundabout self-reference that [...] tosses the monkey wrench into all attempts to prove the prediction unsound."

See also "The surprise examination or unexpected hanging paradox." Timothy Y. Chow. Amer. Math. Monthly 105 (1998) 41-51, a pdf version of which is here.

Solution 3:

All depends on the definition of "surprise exam."

If the teacher states that an exam will definitely be given such that on any morning of the term through the last day students could never know with certainty an exam was scheduled that day, the teacher has spoken falsely, since, if an exam hadn't been given by the penultimate day students would know with certainty the exam was on the last day.

If, though, the teacher states that an unannounced exam will definitely be given at some point in the term, it seems fair to call that a "surprise exam", since only the final day of the class could be predicted with certainty to have an exam if all the others hadn't. And even then, the certainty of the exam would only be known for 24 hrs (not much time to study an entire term's worth of material). All other days would have significant uncertainty. The teacher's strategy to keep students on their toes would work.

So, the paradox of your question comes when you say, "Mathematically it looks like it should be, but that would imply that surprise exams are not possible (and they are)." Surprise exams of the first type are not possible. Surprise exams of the second type are. When you clarify the definitions, there is no paradox.

Solution 4:

If the student is sufficiently bright to prove that a "surprise" exam is impossible then it would certainly be a surprise to the student, whenever the exam was set.

Solution 5:

There are many ways of formalizing the paradox, in many different fields, and the article (by T. Chow) cited by Joel Reyes Noche, does a good review work.

However, one thing which has always bothered me, is that in most formalizations, the surprise exam can take place in the last day, as legitimately as in any other day. To my mind, this doesn't convey the intuitive meaning of "surprise".

A new approach, given by Ran Raz here, doesn't suffer from this fallacy. He follows the "standard" logic formalization - in which "surprise" means "the exact day cannot be proved in advance (using the statement), from the fact it hasn't occured yet", but adds the clause "or it can be proved that it falls on different days" (since, obviously, if opposing facts can be proved, it is hard to say that the students "know" anything). Now, the interesting thing is, that the exam can't occur on the last day, but the induction argument fails from the unprovability of the consistency of the logical system (a.k.a. "Godel's Second Incompleteness Theorem").

I find this approach interesting and refreshing.