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The trio (Posted on 2017-08-21) Difficulty: 3 of 5
You meet a trio of people, A, B, and C. You know that one of them is a knight, one is a liar, and one is a normal, but you are not sure which is which. Normals tell the truth and lie at random. You are allowed to ask three yes-or-no questions. What three questions can you ask to find out which is which?

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Solution solution Comment 1 of 1
Ask A what he would answer if asked if C were normal.

1. If he says yes, then ask B what he would answer if asked if C were normal.  If this is also yes, then it's confirmed: C is normal, as either A or B has to be either a liar or a knight, and in either case is indirectly telling the truth. Then ask A what he would say if asked whether he is a Liar. If he says yes then ABC are LKN respectively; if no, then KLN.

If, on the other hand, on the second question, the one to B, were no, then we know that either A or B is the normal. Actually, we know A is the normal (for "lying") and B was indirectly telling the truth. Now ask C what he would answer if asked if B was a liar; if yes, then the order is NLK; if no, then NKL.


2. If on the other hand A's answer to the first question was no, A is either a liar or knight telling (indirectly) the truth, or the normal happening to tell the truth.  Either way C is not normal and can be relied upon to tell the truth, even if indirectly.

The remaining possible orders are LNK, KNL, NLK, NKL.

So in this case direct your second question to C: If asked, would you say you were a liar?
Then ask him: If asked would you say A is normal? The answers to those questions determine which of the four possibilities is the case.


Summary:

In the descriptions below, I give for the three questions the following format, where x and y are parts of the question "If I were to ask you, is x a y":

Addressee : xy ? answer

After these three questions, with their answers, is the conclusion in the order ABC.

A:CN?Y B:CN?Y A:AL?Y    LKN 
A:CN?Y B:CN?Y A:AL?N    KLN 

A:CN?Y B:CN?N C:BL?Y    NLK 
A:CN?Y B:CN?N C:BL?N    NKL 

A:CN?N C:CL?Y C:AN?Y    NKL
A:CN?N C:CL?Y C:AN?N    KNL
A:CN?N C:CL?N C:AN?Y    NLK
A:CN?N C:CL?N C:AN?N    LNK

Note that NKL and NLK are each represented twice, as A, being normal, decided different ways of answering the first question.

  Posted by Charlie on 2017-08-21 12:23:07
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