perplexus.info :: Liars and Knights : Island President

Island President (Posted on 2012-05-15)

Abe, Ben and Cal live on a remote island inhabited by three groups: the Knights, the Liars and the Knaves. Precisely one of them is a Knight, who always tells the truth, another is a Liar, who always lies, and the remaining one is a Knave, who alternately tells the truth and a lie.

One of the three men is the President of the island.

Abe says:
(1) “The President belongs to a different group from each of the other two of us".
(2) “Ben is not the President".

Ben says:
(1) "The President is a Liar".
(2) "Abe is not the President”.

Cal says:
(1) "Exactly two of us belong to the same group".
(2) "I am not the President".

Who is the President?

See The Solution Submitted by K Sengupta

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Why an island? (spoiler)

| Comment 1 of 2

A1 is True, so Abe is not a Liar.

C2 is False, so Cal is not a knight.

Which leaves just 3 possibilities

A B C President

------- -------- -------- ------------

Knight Knave Liar C, based on C2

Knight Liar Knave A, based on B2

Knave Knight Liar C, based on C2

The president is A or C, so A2 is true, so A is a Knight

And the first possibility is a contradiction, because if C is a lying president, then both of B's statements are true, which makes B a Knight.

So the only possibility is:

A B C President

------- -------- -------- ------------

Knight Liar Knave A, based on B2

And this is internally consistent:

Abe says:
(T) “The President belongs to a different group from each of the other two of us".
(T) “Ben is not the President".

Ben says:
(F) "The President is a Liar".
(F) "Abe is not the President”.

Cal says:
(F) "Exactly two of us belong to the same group".
(T) "I am not the President".

Posted by Steve Herman on 2012-05-15 17:19:55

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