Six friends Andy, Ben, Carl, Danny, Ethan and Frank were hanging out when Frank spotted a wallet lying on the floor. Frank does not have a good memory, but he remembers that the wallet was owned by one of his five friends.

The following are the statement made by the six friends:

Frank: Whose wallet is this?
Andy: That's mine.
Danny: You're lying, Andy.
Carl: Yeah, Andy, you are lying.
Ben: Andy is not lying, guys.
Danny: That wallet is owned by Carl.
Ethan: Yeah, Danny is telling the truth.
Ben: No, Ethan. That's your wallet.

Frank remembers that at least 2 of his friends are the knights who always tell the truth while at least 2 of them are the liars who always lie.
However, Frank cannot recall which of his friends lies or tells the truth.

Can you help Frank to determine who actually owns the wallet?

I solved this two ways.  First I assumed everyone was either a Knight or a Liar.  When I was done, I realized that the condition that there are at least two Knights and two Liars was entirely unnecessary, so then I wondered if the problem intended for the possibility of Knaves (who alternate lies and truths) and solved it that way instead.  Fortunately both interpretations lead to the same answer!

1) Assuming everyone is either a Knight or a Liar:

Consider Ben's two statements: He says that Andy is not lying (about the allet belonging to Andy), and also says that the wallet is Ethan's.  This is obviously a contradiction, so Ben is a Liar, which also implies that Andy is a Liar.

Now consider Danny's two statements: He says that Andy is lying, and that the wallet is Carl's.  His first statement is true, and thus his second must be also. 

Therefore the wallet belongs to Carl.

2) Assuming there might also be Knaves present:

If the wallet belongs to Andy, then Andy is the only one present who could be a Knight, which violates the condition that there are at least two Knights.

Since we know it doesn't belong to Andy, then we know Danny's first statement is the truth.  If his second statement is a lie (i.e. the wallet belongs to Ben, Danny, or Ethan), then the only person present who could be a Knight is Carl.  Everyone else is either a Liar or a Knave, which once again violates the condition that there are at least two Knights. 

Therefore the wallet belongs to Carl.

Posted by tomarken on 2014-04-24 15:57:03