When you are trying to get to Truth Town, you get to No Knaves Town, a city comprised of only Liars and Knights. Each of the three forks leading away from the town (not including the one you came from) leads to a different city.

There are 6 people around. Wanting to know who is what so you can ask them which fork to take, you ask them who is a liar and who is a knight.

A: C would say that B is a liar
B: D would say that C is a knight
C: E would say that F is a liar
D: C would say that A is a knight

Tired of these responses, you ask what fork leads to Truth Town.

E: The left fork leads to Truth Town
F: The middle fork leads to Truth Town

Which fork should you take to get to Truth Town?

My solution has 2 possible outcomes for who is lying & who isn't, but both solutions lead to taking the Right fork to Truth Town.

Assumption #1: F is a liar, E is a knight, L fork goes to TT.
C: E says F is a liar.  True statement.  Thus C is a knight.
D: C says A is a knight.  Premise 1: assume D is a knight.  That makes A a knight as well.
A: C says B is liar.  If A & C are knights, B must be a liar.
B: D says C is a knight.  True statement, but B is a liar.  CONTRADICTION.
Backup 3 steps.
D: C says A is a knight.  Premise 2: assume D is a liar.  "C says A is a knight" is a false statement, so A must be a liar.
A: C says B is a liar.  If A is a liar, B must be a knight.
B: D says C is a knight.  C is a knight, but D is a liar and wouldn't say that.  So B's statement is false even though he is a kngiht.  CONTRADICTION.
Thus, Assumption #1 is false.  The L fork does not lead to TT.

Assumption #2: E is a liar, F is a knight, M fork goes to TT.
C: E says F is a liar.  True statement, thus C is a knight.
At this point, we are in the same situation as we were with Assumption #1.  C is a knight, and the rest will fall out the same.  I won't repeat it all, but we get the same two contradictions and can conclude that the Middle fork does not lead to TT.

We could stop here and conclude that the Right fork is correct, but we'll write it out for fun.

Assumption #3: E & F are both liars, R fork goes to TT.
C: E says F a liar.  False, since E would lie.  Thus, C is a liar.
B: D says C is a knight.  For Premise 1, assume B is a knight and the statement is true.  That makes D a liar.
D: C says A is a knight.  Since D's statement is false, A is a knight.
A: C says B is a liar.  True statement.
No contradictions here, now let's backup 3 steps.
B: D says C is a knight.  Preimse 2: assume B is a liar and the statement is false.  Thus D is a knight.
A: C says B is a liar.  False statement, as C would lie.  Thus A is a liar.
D: C says A is a knight.  True statement, as C would lie.
No contradictions here either.

So while there are two combinations of liars & knights which produce the solution, there is only one answer to the question of which road leads to Truth Town, and the answer is, the Right Fork.

Posted by Audrey on 2004-06-18 16:54:44