The Librarian Paradox
Though I'm sure this isn't something new for many of you, The Librarian Paradox is new to me:
A librarian is wandering round her library one day and comes across a shelf of catalogues. There are catalogues of novels, poems, essays and so on, and some of these catalogues, she discovers, list themselves, while others do not.
In order to simplify the system, the hard-working (and rigorously logical) librarian makes two more catalogues. One lists all those catalogues that list themselves; the other lists all those that don't. Once she has completed this task, she has a problem: should the catalogue which lists all the other catalogues which do not list themselves, be listed in itself? If it is listed, then by definition it should not be listed. However, if it is not listed, then by definition it should be.