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A Fourth Person Problem (Posted on 2006-11-01) Difficulty: 2 of 5
We know that Liars always lie about everything, Knights always tell the truth. and Knaves strictly alternate between lying and telling the truth. All the inhabitants of Island T are Knights, Knaves or Liars.

A visiting tourist was busy in conversation with A, B and C who are inhabitants of Island T, when a fourth inhabitant passed them by. It is known that one of A, B, and C is a Knight; the other is a Knave while the remaining one is a Liar. Nothing definite is known about the fourth inhabitant. A, B, and C, say:

A's statements:
1. The fourth person is a Knight like me.
2. Both B and C have been known to speak falsely.
3. C is less truthful than B or myself.

B's statements:
1. The fourth person is a Knave.
2. He (the fourth person) is not like me.

C's statements:
1. If you were to ask the fourth person, he could claim to be a Liar.
2. The fourth person is a Knight.

Out of the first three, who is the Knight, who is the Liar, and who is the Knave? And what is the fourth person?

See The Solution Submitted by K Sengupta    
Rating: 3.1667 (6 votes)

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Solution Approach 2 | Comment 12 of 16 |

I think I screwed something up in my first approach, so Iím going to start over. Notice I will not rely on details of the wishy washy statements that have been debated here (like A3 and C1).

A1 and A3 must be the same "trueness" no matter what kind of inhabitant she is. If A1 and A3 are both true, then A is either a knight or a knave. If A1 and A3 are both false, then A is either a liar or a knave.

Letís look at the actual statements. Letís assume A1 and A3 are both true. A1 means that A and D are both Knights. That means C2 is true. Since a liar can never tell the truth, and A is already a knight, that means C is a knave, which leaves B to be a liar.

Now letís see if all the statements hold up. We should have A1,2,3 are all true, B1,2 are false, and C1 is false and C2 is true. Uh oh, look at B2. B is a liar and D is a knight, wich means D is not like B. That makes B2 true, which is a contradiction with the fact that B is a liar.

Therefore A1 and A3 are both false.

This means A cannot be a knight.

C cannot be a knight either. If we were, both C1 and C2 would have to be true. That would mean D is a knight AND D could claim to be a liar. But knights can never say "I am a liar" so C1 and C2 contradict each other if C is a knight. Therefore C is not a knight.

Therefore B is a Knight.

Therefore D is a Knave, per B1.

A2 is false because B is a knight and can never have spoken falsely. That means all A1, A2 and A3 are false. Therefore A is a Liar.

Therefore C is a Knave.

I went back and checked each statement against this solution to make sure there still werenít any contradictions. The only one that is still wishy washy is A3 due to the whole "meaning of OR" discussion thatís been going on.

  Posted by nikki on 2006-11-08 12:38:41
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