Liar - Truth-Sayer - Tourist Problem. Construct the answer with the given 2 sub-statements.

Question

A tourist A comes to a country where people are divided into two categories: Liars (L) and Truth Sayers (T). Ls always lie and Ts always speak the truth. Intending to walk to the capital, the tourist comes to a cross road from where the left road goes to the capital. An inhabitant B of the country is standing at the junction.

A asks a single yes/no question of the form “Is it true that _?” to B, and the answer reveals the correct direction to the capital. Construct A’s question by filling in the blank with a statement which makes use of the following two sub-statements (which are either true or false)

(i) Mr. B always lies

(ii) The left road leads to the capital

Answer 1

The goal is to have it so both T and L give the same answer to your question, and what that answer is reveals whether the left road leads to the capital.

T always answers true, so statement (i) will be false. So when speaking to a T, you want $$0 ? (ii) = (ii)$$

When speaking to an L, statement (i) will be true. So you want $$\lnot \left(1 ? (ii)\right) = (ii)$$

You need the answer to change based on (ii), regardless of if statement (i) is true or false. You need it to be false for $(0 ? 0)$ and $(1 ? 1)$, and true for $(0 ? 1)$ and $(1 ? 0)$.

That operation is XOR, also known as exclusive or. When dealing with everyday situations rather than formal logic, the form "or" takes is typically logical XOR. But to avoid ambiguity, you could make the statement as follows.

Is it true that exactly one of the following two statements is true: Mr. B always lies, the left road leads to the capital.

Answer 2

Answer One: $1 \times 1 = (-1) \times (-1) = 1$

You can say: "Is it true that yesterday you would have said that 'The left road leads to the capital'?".

If B is a truth-teller he obviously tells you same as he would have yesterday - the truth.

If B is a liar, then yesterday he would also have lied, and today must tell you the truth (but only about this special case). If he says "Yes", take the leftward road.

Answer Two: $T \oplus F = F \oplus T = F$

You can say: "Is it true that (You always lie) XOR (The left road leads to the capital?)" where XOR ($\oplus$, or sometimes $\veebar$) is the exclusive disjunction. Take the rightward road if Mr. B says "Yes".

Answer 3

For simplicities sake, let's just assume that the left road does indeed lead to the capital. Inhabitant B is either a truth teller or a liar, which will prove useful when constructing an optimal question.

First, let's imagine an alternate scenario: two inhabitants $A$ and $B$, one of whom is a truther and one of whom is a liar. You are allowed only one question, and you may only ask it to a single inhabitant of your choosing. These stipulations create an identical scenario to the original one: You ask one question to one inhabitant who may lie or may tell the truth.

To find success, it is useful to find questions where the answer is not affected by whether the person being asked is a liar or truth-teller. One working category of questions is when you ask about the opposite group of people. A liar would falsely claim that a truth-teller lies, and similarly, and truth-teller would accurately accuse a liar. A question based around these principles would thereby give a consistent answer regardless of who you asked.

One such question is as follows:

$Where\ would\ your\ neighbor\ (the\ other\ person)\ tell\ me\ the\ capital\ is?$

A liar knows that the truth-teller would point you to the left, and would therefore send you right. A truth-teller knows that the liar would misdirect you, and would tell you the rightmost road as well. So no matter who you asked, your answer would be the wrong road.

Taking this back to the single inhabitant gives a slightly modified question:

$Which\ road\ would\ someone\ of\ the\ group\ you\ are\ not\ in\ tell\ me\ to\ travel?$

With the same logic as above, you will always be directed away from the capital. Therefore to find the capital, ask the inhabitant the specified question, and then go the opposite way.