First amthrax came up with the river-crossing riddle I posted here two days ago; I not only solved it, but came up with a proof for why the given solution is optimal.
Now, by chance I've stumbled upon the PartiallyClips strip (LJ: partiallyclips) named "Paradox Dragon". And the logic circuits of my brain kicked in.
Right off the bat, the third panel contains a fairly obvious paradox. (Warm-up puzzle: What?) I'm going to assume that it was put in there intentionally, to give the strip its title; or maybe it simply drifted in as part of the punchline. However, if you discard the third panel entirely ... the first two panels are a self-consistent logic puzzle. I've reprinted the dialogue below, modified to clarify it for problem-solving purposes:
NARRATOR: On the hundredth day of my quest, I faced the Paradox Dragon. I knew that one of its three heads always lied, one always told the truth, and the other head would alternate [back and forth between telling the truth and lying].
FIRST HEAD: You may ask us one question, then you must guess which head is which.
SECOND HEAD: He's lying. You get three questions.
THIRD HEAD: Oh no, it's definitely one question.
NARRATOR: Not willing to risk counting on three questions, I asked [the dragon] ... what the second head would say if I were to ask it if the third head had been lying when it agreed with the first head that I could only ask one question.
FIRST HEAD: (*non-answer*) Oh ... okay, hold on a minute. ...
SECOND HEAD: He'd say, "Yes, the third head was lying."
THIRD HEAD: ... [The second head is] lying [in the statement he just gave].
To my delight, I discovered that the information given was sufficient to triumph over the Paradox Dragon! Anyone else out there care to reproduce my work and solve Rob Balder's riddle?
(n.b.: Comments for river crossing riddles now unscreened. Comments here, as with my other riddle posts, are screened to give everyone time to play along at home.)
Edit, 1-11-2005: Going back through old entries to tag everything up, realized that I never unscreened anyone's solution. Oops! Fixing that now -- keep in mind all answers below were the product of independent work.