The most difficult rstinot in the world: the riddle of the three of the

THE’riddle of the three of the It has been called “the logical puzzle more difficult in the world“Knights and Brigands”, Those logical problems in which a part of the population says Always there truth and the other mind Always. The most famous of these puzzles is “The riddle of the two doors”, also known for having appeared in the film Labyrinth.

The enigma of the three of the adds an additional level of difficulty: in addition to a god who always says the truth And a That mind always, there is also a third That respond in way completely random. And this complicates the situation not a little.

In this article we see the puzzle and one of its solutions.

The riddle of the three of the: between true, false and random

You and your spacecraft have fallen on a mysterious planet. The only way to return to house It is to obtain the help of Three divinities Local: Sincere, liar and random. The gods are willing to help you, but on one condition: you will have to identify correctly each of them, that is, understand which God is sincere, as a liar and which random and therefore assign everyone to each name. But there is a problem: You don’t know who it is who.

It is granted to you Three questions In total, each revolt To one god at a time. The rules are:

• The God Sincere He always says the truth.
• The God Liar mind Always.
• The God Random responds completely randomregardless of the question.

To further complicate things, you can only ask questions whose answer be “yes” or “no”. The gods, however, will not use these words to answer you. In their place they will use “ja” And “from“, Without specifying which means” yes “and what” no “.

How will you do, with only three questions that you don’t understand the answer, to discover the identity of each God?

The solution to the trigger of the three of the

At first glance, this puzzle seems impossible … not for nothing is the most difficult in the world! But, with a little strategy, it can be resolved. The secret lies in carefully formulating our questions. There are several ways to do it, let’s see the simplest one. Let’s start by making a consideration on how to interpret any response is given to us sincere or a liar, the two of the two who do not respond randomly.

How to interpret the answers of sincere and liar: the mechanism of the double lie

Given that We do not know the meaning Of “ja” and “da”, we must structure questions well so that the answer gives us information. Let’s imagine, for example, to want to know from the gods 1+1 ago 2. If we formulated the question thus, however, we would not derive No information, Because we would receive as a response only “ja” or “da”, but we do not know what they mean and we don’t even know with which God we are talking! If, on the other hand, we ask for it like this:

“If I asked you if 1+1 ago 2, would you answer” Ja “?”

We will have useful information. Let’s see why.

Let’s imagine for example that “ja“It means”Yes“Then the whole question becomes:” If I asked you if 1+1 ago 2, would you answer “Yes”? “. In this case the answers of the two of the two would be:

• Sincerewho always tells the truth, would answer us “yes” to the question “1+1 ago 2”, and therefore there replies “yes” also tofull question.
• Liarwhat do I always lie, to the question “1+1 ago 2?” He would answer “no” since he lies. But he has to answer the question “If I asked you if 1+1 from 2, would you answer” yes “?”, And if he answered “no”, he would say the truth about what he would answer! And therefore having to lie also on the entire response, a sort of “double lie” is created for which liar will have to lie about his lie, thus responding “yes”. He too, therefore, replies “yes”.

In both cases, if “Ja” means “yes”, be Sincere That Liar they would answer “ja“And the same identical reasoning we can do it in the event that” Ja “means” no “!

So, whether “ja” means yes, whether it means no, Sincere and liar respond “ja” To the question: “If you ask you if 1+1 ago 2, would you answer” Ja “?”. This because the answer Not Depends from the request main, but from that “internal“(That is,” 1+1 ago 2? “). If the internal question has positive responseboth sincere and liar will answer “ja”. If he has negative responsewill reply “from“. In essence, the meaning exact of “ja” And “from” It does not matter: what matters is the truth or falsity of the phrase contained in the question. The mechanism works because the double lies cancel each other and lead to a coherent response.

Now that we understand how to interpret the answers of sincere and liar, let’s face one of the gods and Let’s start our Three questions.

First question: “If I asked you if the god on my left is random, would you answer” Ja “?”

The first thing we have to try to do is find a god that is not random, so that we can get some information with the second and third question using the technique we saw before. So now, let’s go from God in the center And we ask:

If I asked you if the God to the my left And Random, You would answer “Ja”?

At this point two possibilities open:

• If the God in the center, the one we are talking to, is Randomhis answer will not be of any usefulness. In this case, however, we will have Sincere And Liar to left and right (or left and right), therefore, if we addressed the second and the third request to any of the gods ai “sides“We would be talking to a God who gives coherent answers and we could get useful information.
• If instead we talked to Sincere or with LiarRandom could be both on the right and left. In this case, if we asked the second and third question to a god to the “sides” we would not have the certainty of having obtained a coherent response. And this is a nice problem! Fortunately, if we spoke sincerely or with a liar, we can take advantage of the first logic to understand where it is random. In fact, the answer, even if expressed with words that we do not understand, will allow us to identify random position. Let’s see how.

What to do according to the answers to the first question

With the first question we asked the central god: “If I asked you if the God on my left is random, would you answer” Ja “?”. In this case, the “internal question“Is:“ The God to mine left And Random“.”

Thanks to what we learned about the double lie in the previous paragraph, we know that if we asked sincere or a liar:

• if they answer us “ja”, Then the internal sentence is real: the god on the left is random;
• if they answer us “from”, Then the internal sentence is false: The god on the left is not random, and therefore random must be found on the right.

If, on the other hand, we asked Random, whether he replied “Ja” or “da”, the god random will not be on the right or left. So, in any case, whether we ask sincere, liar or random:

• if they answer us “ja“, Definitely the God a right Not And Random.
• If the answer is “from“, Definitely the God a left Not And Random.

So, even without knowing who mind, who says the truth, or what they mean “ja” and “from”, we can start understanding something.

Second question: “If you ask you” Are you sincere? “, Would you answer” Ja “?”

Now we turn to God that, thanks to the first question, we are sure Not is Random. We ask him:

If I asked you “Are you sincere? “you would answer “Ja”?

As we have seen, if the answer is “ja“, The internal phrase (” Are you sincere? “) Is true, so we are talking to Sincere. If instead he replies “from”, The internal phrase is false: we are talking to Liar.

Third question: “If I asked you” is the one in the center? “, Would you answer” Ja “?”

At this point, we ask God himself, whose identity we now know with certainty:

If I asked you “the one al center And Random? “, Would you answer “Ja”?

Based on his answer, now we can know with certainty the identity of all the gods:

• If with the second question we discovered that we are talking sincerely, if he answers us “ja“We know that the god al center will be Random and theAnother liar. If, on the other hand, with the second question we found Liarthen the God in the center is still random, but the other will be Sincere.
• If he replies “from“And we know from the second question we are talking to Sincerewe know that the god al center It will not be random – and therefore it is Liar – And the other will be random. If, on the other hand, we know we speak with a liar, then, to center we will have Sincere And Random will be the last god left.

In all cases, after this third question we have identified with certainty All three godswithout even knowing what they mean “Ja” and “da”!