# Liars in a Country

A certain country is inhabited by only by people that tell the truth or who tell lies and who respond to questions with yes or no.

A tourist comes to a fork in the road where one branch leads to the capital and the other does not. There is no sign indicating which branch to take, but there is an inhabitant, Mr. Z, standing at the fork. What single question should the tourist ask to determine which branch to take.

This is the situation:

The tourist walks up to the person at the fork in the road who maybe a truth teller or a liar. (Liars always tell lies and truth tellers always say the truth).

Either the left fork leads to the the capitol, or the right fork leads to the capitol.

Either the person (Mr z) is a truth teller or a liar.

So What are the possibilities.

• The person maybe a truth teller (TT) and the left road (LEFT) leads to the capitol.
• The person maybe a liar (LIAR) and the left road (LEFT) leads to the capitol.
• The person maybe a truth teller (TT) and the right road (RIGHT) leads to the capitol.
• The person maybe a liar (LIAR) and the and the right road (RIGHT) leads to the capitol.

So there are 4 possibilities and a single question with a yes or no answer should take care of all theses things.

Because the tourist does not know if the man sitting at the fork of the road is a truth teller or a liar there are four possibilities like this:

The left road may leat to the capital or the right road may lead to the capital.

The man at the fork is a liar or a truth teller.

Now, if the answer is yes, the tourist has to take the left road,

If the answer is no he has to take the left road.

But because the liar always lies, if the answer is yes, he will say no, and if the answer is no, he will say yes. So the correct answers will look like this.

So the question that the tourist needs to ask will be something like this:

“Is it true that you are a truth teller and the left road leads to the capital or  you are a liar and the right road leads to the capital.” This is a single yes or no answer question.

This question is a combination of the two yes cases in the above picture.

If Mr.z is a truth teller and he says yes, then the tourist can take the left road.

If Mr.Z is a liar and the right road leads to the capital then the correct answer will be yes, but Mr.Z will say no so the tourist can take the right road.

Another way of asking the question could be this:

“Is it true, If I ask you whether the left road leads to the capital, you will say yes, won’t you?”

So again you look at the four possibilities : If the man is a truth teller, then the left road leads to the capital, if he is a liar then the right road leads to the capital.

If the answer is yes then the man must be a truth teller and the left road must lead to the capital. The mans answer will be a combination of yes and yes.  If I ask you whether the left road leads to the capital, you will say yes = YES and won’t you? = YES. In this case the tourist will take the left road.

If he is a liar then he has to answer the first part of the question: “If I ask you whether the left road leads to the capital, you will say yes” he will have to answer NO because he is a liar – and he will also have to answer the second part of the question: “won’t you” with NO again, because he is a liar. In this case the Tourist will take the right road.