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You are in a room with three doors. You find out that behind two of these doors, the darkest pit of hell is waiting for you to make a mistake. The other door leads to heaven, where you obviously want to go.

Each door is guarded by a guard:

  • Michael, who tells truth with 75% chance;
  • Vlad, who lies with 90% chance;
  • John, who lies with 70% chance.

You do not know who is who or which door he guards. You may ask each guard 2 questions max but no more than 4 questions in total, because those guys do not like long conversations.

The other thing you have is a magic stone that can be used only once. This stone makes the event with the lowest chance to occur.

You cannot use the stone to do this with multiple events or with an event that has a few independent probabilities. (You cannot ask 2 of the guards at once using the stone). Also, the stone does not count random events.

What is the easiest way which gives you the most chances to go to heaven?

Hint: Be as quiet as possible

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    $\begingroup$ Are only yes/no questions allowed? $\endgroup$ – EKons Apr 14 at 18:56
  • $\begingroup$ @EKons, you may ask them whatever question you want, but there is no guard, who always tells the truth, which makes it difficult to find the correct door using qs like "What's 1 + 1" $\endgroup$ – Andrii Chumakov Apr 14 at 18:59
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    $\begingroup$ Michael, who tells truth with 75% chance; -- What does this even mean? That he tells the truth in 3 out of 4 situations? Or that he always believes he tells the truth, but it might not be the correct answer? $\endgroup$ – Pod Apr 15 at 9:38
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    $\begingroup$ Your problem is based on a false premise. I'd rather drink beers in hell compared to being bored in clouds. $\endgroup$ – Overmind Apr 16 at 7:09
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    $\begingroup$ If I beat myself to death with the stone figuring that I am in some kind of alternate dimension figuring I will wake up in reality then what happens? $\endgroup$ – cybernard Apr 16 at 15:14

11 Answers 11

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No questions are required!

Use the magic stone, then try one of the doors. It's less likely to get the right door (1/3) than one of the wrong ones (2/3), so you'll end up in the right place

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    $\begingroup$ With this answer, I now feel like the puzzle was somehow like a troll puzzle, that tries to distract you from the actual answer. :D $\endgroup$ – Chen Li Yong Apr 15 at 4:23
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    $\begingroup$ Yes, that's the simplest solution, to this problem. The problem is much easier than it seems to be. $\endgroup$ – Andrii Chumakov Apr 15 at 5:33
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    $\begingroup$ I was thinking about this, but I always tought that "yes, but the least likely situation is that you get hit by a bus or turn into a racoon or something else" :D $\endgroup$ – Annosz Apr 15 at 10:20
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    $\begingroup$ But the door is either to heaven or to hell. This isn't a probabilistic thing. $\endgroup$ – Acccumulation Apr 15 at 17:24
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    $\begingroup$ @Accumulation But your choice of the door is a probabilistic thing. $\endgroup$ – Ross Presser Apr 15 at 20:01
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I will ask using the stone: "what door would you tell me that will take me to heaven if I ask you using the stone?". The answer is always the correct door.

For clarify:

If I ask with the stone to Michael, he will be lying. If I ask him (with the stone) "what door will take me to heaven", he will say one of the hell's doors, but I ask about this question (with the stone), so he have to lie and say the heaven's door, because is the only answer that can't be truth. Obviously, Vlad and John will say the truth, so they will say the heaven´s door.

No matter who you ask, he will answer the correct door.

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  • $\begingroup$ You do not know who is who, that's the main problem. Nice try. $\endgroup$ – Andrii Chumakov Apr 16 at 18:17
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    $\begingroup$ @Andrii If I understand Hermes' answer correctly all three will give the same answer, so it doesn't really matter who is who $\endgroup$ – Vincent Apr 17 at 8:47
  • $\begingroup$ The stone can be used only once. You can't ask all three guards with it. $\endgroup$ – Daniel P Apr 17 at 16:20
  • $\begingroup$ I'm not asking all three: only one, at random. Whoever he is, the answer will be the same. $\endgroup$ – Hermes Apr 17 at 16:22
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Ask the following question of all three guards:

If I asked you which door you were guarding, would you say it was the door to heaven?

Now the number of Yeses (Y) will be between 0 and 3 inclusive.

If Y=1, go through that door. The position may either be

1.

(Michael, John, Vlad) = (Yes, No, No)

in which case you go to heaven, or it may be one of

(No, Yes, No) and(No, No, Yes)

in which case you go to hell.

If Y=2, namely

3.

(Yes, Yes, No), (Yes, No, Yes), or (No, Yes, Yes).

then pick one of the Yeses at random and ask the utterer the same question again. If he changes to a No,

go through the door which had the other Yes.

If he doesn't,

pick one of the Yes doors at random and go through it.

If Y=3, namely

(Yes, Yes, Yes)

then again, pick one of the Yeses at random and ask the utterer the same question again.

The number of remaining Yeses will be two or three. Pick one of them at random and go through the corresponding door.

The last possibility is that Y=0:

5.

(No, No, No)

Oh dear.

Pick a guard at random and repeat the question. The number of Yeses will now be one or zero. If it is one, go through the corresponding door. If it is zero, pick one of three doors at random and hope for the best.

This is ignoring the magic stone. I think StephenTG got the right answer. This was basically a trick question. But if you ignore the magic stone, the resulting question is still a good puzzle, and the above may be the best strategy.

Edit

I've edited this in light of Amorydai's helpful comment.

For the three equiprobable cases of which door leads to heaven, the probabilities of each guard saying "Yes" are as follows (writing M, J, V for Michael, John, Vlad):

Case 1: M's door leads to heaven
M: $0.75^2 + 0.25^2 = 0.625$;
J: $2 * 0.7*0.3 = 0.42$
V: $2 * 0.9*0.1 = 0.18$

Case 2: J's door leads to heaven
M: $2 * 0.75 * 0.25 = 0.375$;
J: $0.7^2 * 0.3^2 = 0.58$
V: $2 * 0.9*0.1 = 0.18$

Case 3: V's door leads to heaven
M: $2 * 0.75 * 0.25 = 0.375$;
J: $2 * 0.7*0.3 = 0.42$
V: $0.9^2 + 0.1^2 = 0.82$

In each case, the number of possible sets of answers to the first 3 questions is 8, of which 5 have $Y\not=1$ and therefore require a 4th question. I haven't yet worked out the probability of going to heaven if you use my suggested strategy, but I assert that it is greater than 1/3.

We can start the calculation as follows.

Case 1
YNN has probability $0.625*0.58*0.82= 0.29725$.
YYN has probability $0.625*0.42*0.82= 0.21525$. The probability that we choose the 2nd guard to put our 4th question to is 1/2, and then the probability of his saying N is 0.58 (heaven) and Y (0.42) (half chance of heaven); and the probability of our choosing the 1st guard is also 1/2, upon which the probability of his saying Y is 0.75 (half chance of heaven) and N (hell) is 0.25. So given YYN our chance of getting to heaven is $0.21525 * \big(0.5*(0.58 +0.5*0.42)+(0.5*0.75)\big)=0.125383125$.
So already, having considered only YYN and YNN, we know that our probability of reaching heaven in Case 1 (door to heaven is guarded by Michael) is at least $0.21525+0.125383125=0.340633125$.
Since this is greater than 1/3, we have proved that in Case 1 the strategy is better than picking a door completely at random and I believe that that is also so in each of Cases 2 and 3 :-)

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    $\begingroup$ I’m a bit confused on how you calculated the probabilities. The question you ask will lead to a yes if the guard tells the truth both times or lies both times (assuming the guard is actually guarding the door to heaven). So I understand Michael’s percentage. For John and Vlad it should really be the same, so .7*.7+.3*.3 for John and .9*.9+.1*.1 for Vlad. These are all over 50% so the most likely response would be 3 yeses. $\endgroup$ – Amorydai Apr 15 at 4:55
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    $\begingroup$ That’s if we assume they’re guarding the door to heaven, if they are not, then that would be the percent chance of them saying no. Since they each have a 1/3 chance of actually guarding the door to heaven and 2/3 chance of guarding the door to hell, the chances of them answering yes to that question are actually all below 50% $\endgroup$ – Amorydai Apr 15 at 5:12
  • $\begingroup$ Hmm yes, thanks. You're right. So running through them, for Yeses we get: (Michael) 1/3 * (0.75^2 + 0.25^2) + 2/3 * (0.75*0.25 + 0.25*0.75) = 0.458333; (John) 1/3 * (0.7 ^2 +0.3*^2) + 2/3 * (0.7*0.3 + 0.3*0.7) = 0.473333; (Vlad) 1/3 * (0.9^2 +0.1*^2) + 2/3 * (0.9*0.1+ 0.1*0.9) = 0.393333. Now I am confused. If we ignore the stone what would be the best strategy? $\endgroup$ – h34 Apr 15 at 10:08
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I'd go with this:

Go to any guard, use the magic stone, and ask the question:
Which of the remaining two guards has a greater chance of lying than speaking the truth?

Special thanks to @Amorydai for their valuable feedback in comments

If it is Michel

Since he will lie, so he'll say
"None"

If it is Vlad

He will point to one guard, who will be John, so the other would be Michel

If it is John

He will point to one guard, who will be Vlad, so the other would be Michel

So

I'll identify Michel who will say the truth 75% of times, and ask him the second question
"Which door leads to heaven"
It is a 75% chance that he'll tell the truth

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    $\begingroup$ Not quite. Without the magic stone all the guards a basically random. If you ask one of them “which guard will lie”, they will not have an answer for you. It’s impossible to tell when someone will lie or tell the truth. That’s like asking “when will the next die roll land on a 3”. Whether the guard is lying or telling the truth, he will not know the answer to that. $\endgroup$ – Amorydai Apr 14 at 22:37
  • $\begingroup$ @Amorydai The guards know is who. See OP's comment here $\endgroup$ – Eagle Apr 17 at 6:32
  • $\begingroup$ He is using the stone for that question, which allows him to locate Michael. Then he asks the one who now he knows that is Michael (without the stone), which gives him a 75% chance. He can even increase the chances by asking the other two questions to Vlad and John, knowing that in both cases they probably lie. $\endgroup$ – Hermes Apr 17 at 7:59
  • $\begingroup$ @Akari It doesn’t matter if the guards know who is who. I know when I roll a die it will land on an even number half the time. And I always tell the truth. If someone asks me “on your next die roll will it be an even number?”. Should I say yes or no? I tell the truth, I don’t predict the future. None of the guards will know who will lie next, so the question cannot be answered. $\endgroup$ – Amorydai Apr 17 at 12:17
  • $\begingroup$ @Amorydai oh okay. Now I understood what you meant. Thanks! I've modified my answer. $\endgroup$ – Eagle Apr 17 at 13:33
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What about

Asking them all at once with the stone?
Since the lowest occurrence chance event will happen. Just ask all of them which way is to heaven and two of them will point to one same door.

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    $\begingroup$ Nice try. Sorry, but the stone can be only used to make one event happen $\endgroup$ – Andrii Chumakov Apr 15 at 17:24
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The real question is:

How do you know that 1 of the doors leads to heaven and the other 2 lead to hell? If you learn that from the guards, that means that the correct door is most likely Vlads door, because he has the highest chance of lying and his door supposedly leads to hell.

Further more:

The text for the stone states the "stone makes the event with the lowest chance to occur". Excluding random events we could come up with, the event with the lowest chance to occur would be Vlad telling the truth.

Thus:

You would use the stone and ask him "Does this door lead to hell." His answer should be "No".

Also:

It fits with the 4 question rule. The first 3 questions are you asking each guard what their name is, Vlad should lie to you that he is John and the final question is the one directed at Vlad with the stone.

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  • $\begingroup$ Vlad can also lie that he is Michael, and there is no correlation between the doors and the guards, and also, the information given is true, there is no need to doubt that. Nice try $\endgroup$ – Andrii Chumakov Apr 16 at 15:11
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You just need one question and u have to use the stone.

Ask the most left guard: "What would the middle guard answer, if I would ask him: What would the right guard answer, if I would ask him what's the door to hell" (crazy question but I needed to include all 3 guards in one question)

With the stone, the event with the lowest chance would be that 2 guards will tell the truth and one will lie. -> The asked guard will lie -> He points on the door to heaven! -> 100% chance to go to heaven!

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Just ask the room

"Which is the door to heaven?" while using the stone. The least likely outcome is that Vlad will answer you truthfully, so whoever answers you is Vlad, and he's telling you the correct door.

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Well, the post by StephenTG is correct without asking questions. Following the hint, here's my answer for if you HAVE to ask a question.

Because Vlad LIES 90% of the time, then he's truthful 10% of the time. Therefore, by using the stone to make that 10% happen, Vlad tells the truth of which door to go through.

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  • $\begingroup$ That would be a great idea but, you don't know who is Vlad. And yes, the questions are not required $\endgroup$ – Andrii Chumakov Apr 18 at 16:36
  • $\begingroup$ Can't you ask all three at once? There's one question, and you can eliminate try to decide who Vlad is. Wait... they may try to get you to think Michael is Vlad, therefore giving a 25% of lying, therefore, the stone makes it a lie... $\endgroup$ – CStafford-14 Apr 18 at 16:41
  • $\begingroup$ Vlad has the highest chance of lying, therefore, the stone will make the least likely to happen: Vlad telling the truth. $\endgroup$ – CStafford-14 Apr 18 at 16:44
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    $\begingroup$ John would behave just like Vlad if you use the stone. Let me explain - there are two misunderstandings about the stone. First of all: the 'least likely' applies for a set of events, excluding random. Playing poker won't result in asteroid falling down. Secondly: In John's case there are two outcomes: telling the truth with 30% chance or telling a lie. Using a stone, will cause the least likely one to occur with 100% chance. Stone can be used to 'correct' one outcome, not a lot. $\endgroup$ – Andrii Chumakov Apr 18 at 16:47
  • $\begingroup$ As long as I don't ask Michael, I'm fine. $\endgroup$ – CStafford-14 Apr 18 at 17:54
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If, contrary to StephenTG's answer, we interpret the stone as only applying to random events, then we can simply ask

Which of these doors leads to hell?

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  • $\begingroup$ You might want to elaborate on your answer. MIchel will show you one door (choose this one) or the other two will show you two doors (choose the last one) $\endgroup$ – Armin Apr 16 at 11:43
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Am I confused...It seems like asking Vlad with the stone is the best option because is has a 10% chance of telling the truth. "This stone makes the event with the lowest chance to occur". So Then Vlad tells the truth, and you leave asking only 1 question?

Then just ask Michael a couple of times for fun because you already know the truth. Michael should agree with Vlad at least once if not more.

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  • $\begingroup$ How do you know who is Vlad and who is not? $\endgroup$ – Eagle Apr 15 at 20:16
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    $\begingroup$ @Akari Hmmm He's the one with the Russian accent. Suppose that would be too easy. $\endgroup$ – cybernard Apr 15 at 20:18
  • $\begingroup$ Are you sure that there can never be any Russians with Michel or John as their names? And that Vlad is definitely Russian? $\endgroup$ – Eagle Apr 15 at 20:19
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    $\begingroup$ @SamyBencherif Jokes on you, it's Vlad the Impaler, thus Romanian/Transylvanian xD $\endgroup$ – Armin Apr 16 at 11:42
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    $\begingroup$ @cybernard, please, do not rely on russian accent to deduce who is who. I'll make an edit, to make sure its clear. Nice try $\endgroup$ – Andrii Chumakov Apr 16 at 15:08

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