...on your part 2 premises. Or at least, I think it's overly complicated.
IN short, what you deduced was this: The problem says the quiz is a surprise. Since the problem says it's a surprise, the student's can't possibly have deduced the day of the quiz.
One might just as easily give a parallel problem:A professor told the students "Today, I will be adding 2 + 2 and informing the class of my results.". Adding 2 + 2 is impossible. What's the problem in the story?
and then follow this chain of logic.
- The problem says adding 2 + 2 is impossible
- Therefore the professor was overly optimistic
I think your first take on the problem is more accurate. Here's another way to look at it:
The professor says it will be a surprise. We're interested in whether or not the professor is mistaken.
At the time the professor makes the statement, clearly it can be a suprise; assuming a constant distribution, there is a 1/5 chance for each day, so the students can't know which day it will occur.
If there is no quiz on Monday, then there is a 1/4 chance for each of the remaining days.
The last day, then, becomes unique. If there is no quiz on Thursday, there is a 1/1 chance of the quiz on Friday. So the students can't get through class on Thursday without knowing when the quiz is.
But this doesn't mean the quiz isn't a surprise -- it merely means the last possible moment for it to be a surprise is Thursday's class. This falls directly out of needing 2 options to have a surprise. Because the last day is unique; using the last day as the seed for an inductive proof is not valid.
However, if the professor had said "The quiz will be a surprise at the moment it is given", things get more interesting. Then it turns into a "This statement is false" kind of paradox.