Problem

You are in a land inhabited by people who either always tell the truth or always tell falsehood. You come to a fork in the road and you need to know which fork leads to the capital. There is a local resident there, but he has time only to reply to one yes-or-no question. What one question should you ask so as to learn which fork to take?

Solution

Suppose we label the two roads of the fork as “Left” and “Right”, or L and R for short. The important condition we use to solve this riddle is the fact that everyone in the land are either all truth tellers or all liars.

Q: If I ask another man does the left road lead to the capital, would he say yes?

Consider the four possible cases of the man replying:

Case 1: He is a truther, and he responds yes

So if we ask another man if the left road leads to the capital, he will say yes. Since everyone in the land is a truth teller, this hypothetical man would be telling the truth. Thus, the road on the left leads to the capital.

Case 2: He is a truther, and he responds no

So if we ask another man if the left road leads to the capital, he will say no. Since everyone in the land is a truth teller, this hypothetical man would be telling the truth. Thus, it is not the road on the left, but the road on the right which leads to the capital.

Case 3: He is a liar, and he responds yes

Since the man is a liar, the truth is that if we ask another man if the left road leads to the capital, he will say no. Since everyone in the land is a liar, this hypothetical man would be lying. Thus, the road on the left would indeed lead to the capital.

Case 4: He is a liar, and he responds no

Since the man is a liar, the truth is that if we ask another man if the left road leads to the capital, he will say yes. Since everyone in the land is a liar, this hypothetical man would be lying. Thus the road on the left does not lead to the capital, so the road on the right does.

Now, suppose we ask the man the question, and he responds yes. In either case, if he is a truther or a liar, this means that the road on the left leads to the capital. On the contrary, if the man replies no, then it must be that the road on the right leads to the capital.