Re: S, B, and a puzzle by Boolos, Smullyan, McCarthy

From: Jesse Mazer <lasermazer.domain.name.hidden>
Date: Tue, 12 Oct 2004 15:13:45 -0400

Bruno Marchal wrote:

>You made a relevant decomposition of the problem,
>and you are on the right track. Actually I'm not sure the "ja" "da"
>McCarthy's amelioration adds anything deep to the problem.
>It will be enough to take into account that a double negation
>gives an affirmation.

I've thought about McCarthy's version for a while and I can't figure it out,
even if I make the simplifying assumption of two truth-tellers and one
random-answerer. It seems like the 3 bits of information you get from their
answers just shouldn't be enough to tell the meaning of Ja vs. Da *and* tell
you the identity of all three gods. 3 bits of information should only allow
you to choose from 8 possibilities, but there seem to 12 possibilities here:

(Ja=yes, god #1=knight, god#2=knave)
(Ja=yes, god#1=knight, god#2=knife)
(Ja=yes, god#1=knave, god#2=knight)
(Ja=yes, god#1=knave, god#2=knife)
(Ja=yes, god#1=knife, god#2=knight)
(Ja=yes, god#1=knife, god#2=knave)
(Da=yes, god #1=knight, god#2=knave)
(Da=yes, god#1=knight, god#2=knife)
(Da=yes, god#1=knave, god#2=knight)
(Da=yes, god#1=knave, god#2=knife)
(Da=yes, god#1=knife, god#2=knight)
(Da=yes, god#1=knife, god#2=knave)

Then I thought there might be a clever solution where you can figure out the
identity of the gods without ever knowing the meaning of "Ja" and "Da". But
if you don't figure out the meaning of "Ja" and "Da", it seems to me you're
really only getting 2 bits of information from their answers, since it
doesn't actually matter what the first answer is, only whether the second
and third answer are the same as or different from the first--you can't
distinguish between the answers Ja-Da-Ja and Da-Ja-Da, for example. 2 bits
should only be enough to choose from 4 possibilities, but there are 6
possibilities you need to choose from to find the identity of all three
gods:

(god #1=knight, god#2=knave)
(god#1=knight, god#2=knife)
(god#1=knave, god#2=knight)
(god#1=knave, god#2=knife)
(god#1=knife, god#2=knight)
(god#1=knife, god#2=knave)

So either way I'm stumped...can you give me the solution? If you want to let
other people keep trying to figure it out you can just email it to me.

Jesse
Received on Tue Oct 12 2004 - 15:23:42 PDT