QUESTIONS AND PROBLEMS 559one of the regular class meetings the following week. One of the brighter students
then reasons as follows. The exam will not be given on Friday, since if it were,
having been told that it would be one of the three days, and not having had it on
Monday or Wednesday, we would know on Thursday that it was to be given on
Friday, and so it wouldn't be a surprise. Therefore it will be given on Monday or
Wednesday. But then, since we know that it can't be given on Friday, it also can't
be given on Wednesday. For if it were, we would know on Tuesday that it was to
be given on Wednesday, and again it wouldn't be a surprise. Therefore it must be
given on Monday, we know that now, and therefore it isn't a surprise. Hence it is
impossible to give a surprise examination next week.
Obviously something is wrong with the student's reasoning, since the instructor
can certainly give a surprise exam. Most students, when trying to explain what is
wrong with the reasoning, are willing to accept the first step. That is, they grant
that it is impossible to give a surprise exam on the last day of an assigned window
of days. Yet they balk at drawing the conclusion that this argument implies that
the originally next-to-last day must thereby become the last day. Notice that, if the
professor had said nothing to the students, it would be possible to give a surprise
exam on the last day of the window, since the students would have no way of
knowing that there was any such window. The conclusion that the exam cannot
be given on Friday therefore does not follow from assuming a surprise exam within
a limited window alone, but rather from these assumptions supplemented by the
following proposition: The students know that the exam is to be a surprise and they
know the window in which it is to be given.
This fact is apparent if you examine the student's reasoning, which is full of
statements about what the students would know. Can they truly know a statement
(even a true statement) if it leads them to a contradiction?
Explain the paradox in your own words, deciding whether the exam would
be a surprise if given on Friday. Can the paradox be avoided by saying that the
conditions under which the exam is promised are true but the students cannot know
that they are true?
How does this puzzle relate to Godel's incompleteness result?