Surprise exam paradox?
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:
- The student doesn't know exactly at which date the exam is at any state.
- 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.
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.
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.