B What Single Question Reveals the Door to Freedom?

[kent davidge]

I saw this problem in Wikipedia in my native language. The problem goes as follows

You are detained and there are two doors in front of you, one leads to death and the other to freedom. Each door is guarded by one deputy. One of the deputies always makes false statements while the other always makes true statements.

You can ask just one question to just one of the deputies (in order to know what door is the one you want to go through). What should you ask?I did the following: as one deputy will always make false statements and the other always true statements, I can ask anyone out to choose one of these options:

(a) I'm a chicken; this door leads to death.
(b) I'm a chicken; this door leads to freedom.
(c) I'm a human; this door leads to death.
(d) I'm a human; this door leads to freedom.

I choose anyone deputy to ask this question and if he chooses (a) or (b) then it's the liar, therefore if he chooses, say, (a) we know that the door leads to freedom... the same reasoning holds for the two bottom options... and in this way I can know what door leads to freedom and what leads to death.

Does this solution sounds logic?

 

[kuruman]

The four options you have are not one question, they are a multiple choice among four statements. If you form these as questions, you will be asking four questions and not one as you are required to do. Try another approach.

 

[jedishrfu]

As @kuruman has said the a thru d choices are yours to make, pick one and then try them against the two deputies to see what door to select.

 

[kent davidge]

It's meant to be one question in the sense that I ask him "please, choose one alternative out of these four".. but yea

pick one and then try them against the two deputies to see what door to select.

if one does that and one gets "I'm a chicken; this door leads to death" one concludes what is the guarded door.

 

[QuantumQuest]

Does this solution sounds logic?

As has already been noted by kuruman you should ask just one question and not four. Now, to give you a hint about the solution try two things: first, it is not meaningful to just ask the question directly - as you don't know who is telling the truth and who's not. This gives a hint that you should ask a question that reveals who is telling the truth so you have to concentrate on a question regarding the guards themselves. Second, think in terms of reversing answers.

 

[PeroK]

It's meant to be one question in the sense that I ask him "please, choose one alternative out of these four".. but yea

if one does that and one gets "I'm a chicken; this door leads to death" one concludes what is the guarded door.

You might like this:

 

[kent davidge]

You might like this:

Cool

you have to concentrate on a question regarding the guards themselves. Second, think in terms of reversing answers.

Maybe a question that I know the answer a priori so that depending on the answer I know who is/is not the liar, and that same answer should also imply the identity of the doors...

 

[PeroK]

Cool

Maybe a question that I know the answer a priori so that depending on the answer I know who is/is not the liar, and that same answer should also imply the identity of the doors...

The question as you quote it has been spoiled by assigning a guard to a door, allowing simple questions about known facts to be used to identify the liar and the door.

The question needs either only one guard or the guards to be dissociated from the doors in order to prevent this.

In that case, you can easily establish who is the liar with one question, but that leaves you still not knowing which door is which.

 

[kuruman]

The question needs either only one guard or the guards to be dissociated from the doors in order to prevent this.

That is as it should be. Here is a hint to think about. It is also assumed that the deputies have worked together long enough to know whether the other guy is a liar or a truth-teller.

 

[phinds]

Just ask either one of them

Spoiler

"If I were to ask you if this is the door to freedom, would you say yes?"

If it's the door to freedom, both will say yes and it it's not, both will say no.

 

[QuantumQuest]

Maybe a question that I know the answer a priori so that depending on the answer I know who is/is not the liar, and that same answer should also imply the identity of the doors...

To help further towards the solution I know - I don't know if there are other solutions, one more hint - also take note of the hint of @kuruman in post #9 because this is a crucial assumption in my solution too: make use of some conditional. This is meant to be asked to one guard, even though you don't know who tells the truth and who is not. If you combine this with my first hint in post #5 it should be easy enough to find the appropriate question. Also the second hint there guides to the formulation of this question.

 

[kent davidge]

The answer according to wikipedia https://pt.wikipedia.org/wiki/Lógica is

(free translation)

"Ask to anyone of them: what door would you partner point out as being the door to freedom?

Explanation: the liar would point out the door of death as being the door that his partner would say is the door of freedom, because it's a lie on the affirmation of the truthteller. And the truthteller, knowing that his partern always lie, would say that he would point out the door of death as being the door of freedom.

Conclusion: either deputy would point out the door of death as being the door that his partner would say is the door of freedom. Therefore, you know you should walk through the other door."I guess this is the kind of question @kuruman and @QuantumQuest were thinking about.

 

[kuruman]

I guess this is the kind of question @kuruman and @QuantumQuest were thinking about.

You guessed correctly, in my case anyway.

 

[QuantumQuest]

The answer according to wikipedia https://pt.wikipedia.org/wiki/Lógica is

(free translation)

"Ask to anyone of them: what door would you partner point out as being the door to freedom?

Explanation: the liar would point out the door of death as being the door that his partner would say is the door of freedom, because it's a lie on the affirmation of the truthteller. And the truthteller, knowing that his partern always lie, would say that he would point out the door of death as being the door of freedom.

Conclusion: either deputy would point out the door of death as being the door that his partner would say is the door of freedom. Therefore, you know you should walk through the other door."

I have not really seen Wikipedia answer but this is a puzzle I have solved about thirty five years ago. It is a well known puzzle - especially in the math world, and if you see my hints indeed this is the solution that I hint towards too. That's why I told you to ask some "if" regarding guards themselves and reverse answer ;)

 

[kent davidge]

I have solved about thirty five years ago

Wow, a lot of time

 

[phinds]

Wow, a lot of time

Not really. I solved it about 60 years ago. This is a VERY old logic/linguistic puzzle. See post #10.

 

[QuantumQuest]

Not really. I solved it about 60 years ago. This is a VERY old logic/linguistic puzzle. See post #10.

Hey I'm not that old to know that ;)

 

[sysprog]

Hint1: You can ask a single yes-or-no question that embeds your question about the door inside a question about the guard.
Hint2: You don't care which guard is which; just which door is which.
Hint3: You can construct the question so that the answer is the same regardless of which guard is asked.

Answer:

Spoiler

You ask either guard, "if I were to ask you, 'does the door you're guarding lead to freedom', would you say yes?"
If the guard says Yes" you walk out that door; if he says "No", you exit by the other door.

Explanation:

Spoiler

You don't simply ask the guard about the door; you instead ask what he would say if asked about the door. That way, you get either a lie about a lie, which comes out as the truth, or the truth about the truth, which also comes out as the truth (just as either a double negative or a double affirmative is an affirmative).

If it's the good door:
the truth teller says "Yes" -- he knows that he would not lie and say "No" if asked if it was the good door, so he tells the truth about that and says "Yes".
the liar also says "Yes" -- he knows that he would lie and say "No" if asked if it was the good door, so he lies about that and says "Yes".

If it's the bad door:
the truth teller says "No" -- he knows that he would not lie and say "Yes" if asked if it was the good door, so he tells the truth about that and says "No".
the liar also says "No" -- he knows that he would lie and say "Yes" if asked if it was the good door, so he lies and about that and says "No".

There are other ways to formulate the question; all of them require nesting so that the liar property is invoked an even number of times.

 

[phinds]

Hint1: You can ask a single yes-or-no question ...

I take it you didn't read post #10

 

[sysprog]

I take it you didn't read post #10

You're right -- you said the same thing -- and more succinctly.

 

[kent davidge]

Hint1: You can ask a single yes-or-no question that embeds your question about the door inside a question about the guard.
Hint2: You don't care which guard is which; just which door is which.
Hint3: You can construct the question so that the answer is the same regardless of which guard is asked.

Answer:

Spoiler

You ask either guard, "if I were to ask you, 'does the door you're guarding lead to freedom', would you say yes?"
If the guard says Yes" you walk out that door; if he says "No", you exit by the other door.

Explanation:

Spoiler

You don't simply ask the guard about the door; you instead ask what he would say if asked about the door. That way, you get either a lie about a lie, which comes out as the truth, or the truth about the truth, which also comes out as the truth (just as either a double negative or a double affirmative is an affirmative).

If it's the good door:
the truth teller says "Yes" -- he knows that he would not lie and say "No" if asked if it was the good door, so he tells the truth about that and says "Yes".
the liar also says "Yes" -- he knows that he would lie and say "No" if asked if it was the good door, so he lies about that and says "Yes".

If it's the bad door:
the truth teller says "No" -- he knows that he would not lie and say "Yes" if asked if it was the good door, so he tells the truth about that and says "No".
the liar also says "No" -- he knows that he would lie and say "Yes" if asked if it was the good door, so he lies and about that and says "No".

There are other ways to formulate the question; all of them require nesting so that the liar property is invoked an even number of times.

Thank you. Your post made it clear and easy to understand the logic behind the question.