Re: Another knight/knave hat puzzle
- From: amorgan@xxxxxxxxxxxxxxxxxx (Alan Morgan)
- Date: Wed, 13 Sep 2006 10:36:22 -0700 (PDT)
In article <Pine.LNX.4.61-042.0609131305230.8539@xxxxxxxxxxxxxxxxxxxxx>,
Arthur J. O'Dwyer <ajonospam@xxxxxxxxxxxxxx> wrote:
On Wed, 13 Sep 2006, Alan Morgan wrote:
Ed Murphy <emurphy42@xxxxxxxxxxxx> wrote:
Personally, I'd enjoy seeing some puzzles involving another of
Smullyan's islands - where each native is either sane or insane,
and is also either a human or a vampire:
A sane creature's beliefs are all true.
An insane creature's beliefs are all false.
A human makes only statements he believes are true.
A vampire makes only statements he believes are false.
ObWarmup: Since sane humans and insane vampires both make true
statements, how can you tell them apart?
Are you sane? Sane humans believe they are sane and tell the
truth. Insane vampires also believe they are sane and lie about
Yes; or "Are you a sane human?". :)
Yes, although continuing with my argument below, why does an insane
vampire have to believe that he is a sane human? Can't he believe
that he is an insane human? He'd obviously have to believe that he's
sane along with that, so he's sane *and* an insane human, which is
contradictory. Of course, he doesn't believe that it is contradictory,
but since he doesn't believe that he believes those things, that's
hardly a problem (the fact that he believes that he believes that he
believes those things either is or isn't a problem depending on how
much you've had to drink).
In one of Smullyan's books,
possibly the one to which you refer below, he makes the joke even more
obvious: Suppose we know of two twin brothers, John and James, who
both respond perfectly truthfully to any question. How can we determine
which one is John and which one is James if we are allowed to ask only
one question of each brother? (Spoiler below.)
My satisfaction at solving that puzzle lasted only a couple of seconds.
I remember having some issues with this when I read about these
puzzles in Smullyan's book _I Can't Recall The Title_ (that was
not the title of the book, but wouldn't it be a great one?).
I bet it was "What is the Name of This Book?"
I thought it was _To Mock A Mockingbird_, but you may be right.
I think it involved vampires and humans wearing hats and I
concluded that an insane vampire wearing a white hat could claim
that his hat was black perfectly easily, but the solution required
that insane vampires in white hats would claim that their hats
were white. At least, I think that was it. You ran into this
problem any time there was more than one possible wrong belief.
Hmm; I never thought of it that way. I believe Smullyan considered
the unique "wrong belief" in this case to be the logical negation:
an insane person wearing a white hat would believe that he was /not/
wearing a white hat!
Would he believe that he is wearing a hat at all?
Naturally, since he's not wearing a black or purple hat, he would
also believe that he was wearing a black hat, and also that he was
wearing a purple hat, which can't both be true; but he's insane,
so it's okay for him to hold mutually incompatible beliefs.
I do see the problem with insane vampires, but hey, it's a puzzle!
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