Three gods, A, B and C, are called Truth, Falsehood and Random, in some order. - Truth always answers with the truth. Falsehood always responds with a lie. Random answers at random. They understand your language, but they respond using the words “da” and “ja,” and you don't know which means “yes” and which means “no.” You can ask exactly three questions, each directed to a single god and with a yes/no answer. How do you identify with certainty which is which?

Yes: it is solved in 3 questions using the control form 'If I asked you P, would you answer ja?' to neutralize lies and language.

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The most difficult logic puzzle in the world

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His fame is not exaggerated. It brings together three different difficulties in a single knot—truth, lies, and unknown language—and demands a solution that neutralizes them at the same time.

Three gods, A, B and C, are called Truth, Falsehood and Random, in some order. - Truth always answers with the truth.

Falsehood always responds with a lie.
Random answers at random. They understand your language, but they respond using the words “da” and “ja,” and you don't know which means “yes” and which means “no.” You can ask exactly three questions, each directed to a single god and with a yes/no answer. How do you identify with certainty which is which?

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Hints

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Before tagging the three, you need to ensure an interlocutor who is not Random.
“If I asked you…” questions neutralize both the lie and the unknown meaning of ja/da.
The first question does not seek a final name, but rather to isolate a reliable god for the next two.

Solution

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Answer: The three gods can be identified in exactly three questions. We will always use the same form of question. For any $P$ proposition, we will ask: > “If I asked you if $P$, and you answered ‘da’, would you be telling me the truth?” Let's call this form $M(P)$. The advantage of $M(P)$ is that, when addressing a non-Random god, your response does not depend on knowing whether “da” means yes or no. It only depends on two things: whether the god is True or False, and whether $P$ is true or false. The table is this:	God asked	$P$ true	$P$ false
Truth	da	ha
Falsehood	ha	da	This table will be the main tool. Now we place the gods in a circle: $$A\to B\to C\to A.$$ For each god $X$, we will call “next” the god who comes after him in that circle and “previous” the one who comes before him. We define the proposition: $$P_X:\ \t\t\t\t\t\t\t\t\text{“el siguiente de }X\t\t\t\t\t\t\t\t\text{ es Verdad o el anterior de }X\t\t\t\t\t\t\t\t\text{ es Falsedad”}.$$ The important word is “or.” It is not an “if and only if.” First question Question to A: > “If I asked you if $P_A$, and you answered ‘da’, would you be telling me the truth?” Now act according to the answer: - if A answers ha, the second question will be for B;

if A answers da, the second question will be for C. Second question Ask the chosen god the same kind of question, but applied to him: - if you ask B, you ask for $P_B$;
if you ask C, you ask for $P_C$. With the two answers, the Random god is located like this: | Answer from A | Second question to | Second answer | Random |

|---|---|---|---|
| ha | B | da | C |
| ha | B | ha | A |
| da | C | da | A |
| da | C | ha | B | The proposition is designed so that, between the two non-random gods, the responses form an ordered pattern: one responds ja and the other responds da. The Random god, on the other hand, can match that pattern or break it. Therefore, when choosing the second question based on the first answer, the two answers are enough to locate you. After these two questions, Random is already identified. Third question There are now two non-random gods left: one is Truth and the other Falsehood. Choose one of them, call it $X$, and ask: > “If I asked you if you weren't Random, and you said 'da', would you be telling me the truth?” Since we already know that $X$ is not Random, the proposition “you are not Random” is true. According to the initial table: - if you answer da, $X$ is True;

if you answer ha, $X$ is Falsehood. The other non-random god is identified by discard. Thus, with the first two questions, Random is located, and with the third, a distinction is made between Truth and Falsehood.

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