# Three doors, one question

A prisoner wakes up in an empty room (being seated on the floor against the wall). There are three solid doors on three of the room's walls: one to his left, one in the middle in front of him and one to his right. The fourth wall behind him) is a thick glass pane with an intercom built in.

Through the intercom the warden says: "One of these doors leads to freedom, the other door [note the singular] leads to death. You may choose one door for me to unlock. You may ask me two questions which I will answer with 'yes' or 'no' to help you choose." (but not "maybe" or any other third value: in that case, the answer will be "no".)

The prisoner is confused and hence asks: "What's behind the third door?" [b]

The warden answers: "No, I can't tell you, because that's not a yes-or-no question. It still counts as your first question though."

The prisoner complains: "That's unfair!"

The warden responds: "No, it isn't, you don't even need that second question to survive."

After some contemplation the prisoner asks his second question, chooses a door and goes free.

1. what question did the prisoner ask and what was the answer?
2. what was behind the third door [a] and what was the probability for the prisoner to pick it?

(If multiple valid solutions the one with lowest probability for the third door wins.)

[a] Hint: as some already noticed, the third door leads back to prison (the prisoner had to be brought into the room in some way, after all), hence leads neither to death nor to freedom.

[b] Note: after to the prisoner's first question, the warden knows that the prisoner doesn't know what is behind the third door - but the warden knows that the third door leads back to the prison.

• No, in that question there are three possible answers ("yes/no/maybe") for three values ("1/2/3"). Here there are only two possible answers ("yes/no") for three values ("left/middle/right"). Commented Mar 10, 2015 at 19:03
• By "third door" the prisoner referred to the fact that there are three doors, but he knows of only two things behind them, freedom and death: so not only doesn't he know what the third thing is, he also doesn't know behind which door it is. So it's about mapping the doors ("left/center/right") onto what is behind them ("freedom/death/third thing"). Commented Mar 10, 2015 at 19:13
• The prisoner could turn towards the camera and speak directly to the viewer, breaking the fourth wall, and thereby escaping. Commented Mar 10, 2015 at 20:19
• @user66554 -1 Ambigous, not a logic puzzle Commented Mar 11, 2015 at 1:04
• Is the problem claiming that the warden asserts that the prisoner can ask "Is the sky blue" and be assured survival with the information he currently has? If so I think the warden is lying and hence the puzzle is unsolvable. Commented Jun 5, 2015 at 12:33

The prisoner asks "I am thinking of either the left or middle door. Assuming that the third door also leads to my death and is, therefore, essentially the same as the death door, is there a door further right of the one I am thinking of that would be better for me to choose?"

If door "L" is correct, the answer will be no. If you are thinking of "L" everything else is worse while if you are thinking of "M" then "R" is no change.

If door "M" is correct, the answer will be "I don't know". If you are thinking of "L" the answer is yes while if you are thinking of "M" then "R" is worse.

If door "R" is correct, the answer will be "yes". If you are thinking of "L" the answer is yes while if you are thinking of "M" then "R" is better.

With the warden now answering "no" to a question where he doesn't know that answer the puzzle changes. I just realised the "survive" wording which means my less rigorous answer is inferior to this one.

The prisoner should ask "Is the Middle door better for me than the Left?".

If the answer is "yes" the middle door is either prison (because the left door is death) or freedom (because the left door is death or prison). The prisoner should pick middle.

If the answer is "no" the left door is either prison (because the middle door is death) or freedom (because the middle door is death or prison). The prisoner should pick left.

This strategy yields a 100% chance to survive. There is a 33.3% chance he will end up in prison and a 66.7% chance he will be freed.

• How can you make the assumption that the third door is equal to the death door? If it leads to freedom or the warden's office then the reasoning will be off, possibly causing you to choose death. Commented Mar 10, 2015 at 19:31
• The prisonner asks the warden to treat the warden's (third) door and the death door as equivalent. Whether there is a difference or not, the warden is not playing fairly if he mistreats this requested assumption. It therefore will only effect the warden's answer. Commented Mar 10, 2015 at 19:43
• Alright. I misunderstood who's supposed to do the assuming. So if he would answer "I don't know" then that'd be the right answer. But since he'd answer "no" instead it would send the prisoner to his death if the middle door were correct. Commented Mar 10, 2015 at 19:46
• Exactly, my answer assumed he wouldnt lie. As he does, it would bite to be the prisoner. Commented Mar 10, 2015 at 19:48
• Well, you aren't giving him any other choice, given that he stated that he can answer only "yes" or "no" but try to ask a question that he cannot answer with "yes" or "no". He was already being nice on the first question by elaborating, but he really can't do that on the second. Commented Mar 10, 2015 at 19:51

ooh I got it.

the best question is

is the left door a worse choice than the middle door?

if yes

pick middle

if no

pick left

that's 6/6 chances of survival and 4/6 chances of freedom

• That's the same answer as that by kaine, except for details in wording. Commented Mar 10, 2015 at 20:53
• ah, I didn't see. since you are not marking either of them as right, should we keep searching? Commented Mar 10, 2015 at 20:56
• I know of one solution better than 4/6 freedom. Maybe there even is one with 100% freedom, but that I don't know. So I'll wait a bit if someone comes up with one. Commented Mar 10, 2015 at 21:05
• @user66554 correct me if I am wrong but in order for the probability of going to jail to be less than 1/3, you must have a non-zero chance to select each of the three doors based on the warden's answer. As the warden can only give two answers (and can't not give an answer), doesn't that mean his answer can only indicate between 2 of the doors? Commented Mar 10, 2015 at 21:24
• sup with the 6/6 4/6 when it's easier to understand with 3/3 2/3 since there are 3 doors. Commented Mar 11, 2015 at 1:40

ok, I think I figured out the missing "trick" that we need to solve this one. it's when the warden says "You may choose one door for me to unlock."

the warden is still in the other room, so to unlock a single door we can presume that he has to walk into the prisoner room himself, by going through one of the 3 doors. so, the answer is to ask the question we figured out earlier "is the left door a worse choice than the middle door?". if yes, pick the middle door, if no, pick the left door. that gives us 4/6 chances freedom and 2/6 chances back to the warden room. the trick is to not tell your choice out loud, but rather tell the warden that you are ready to pick a door. if the walks into the room through the same door you had picked, choose the rightmost door instead.

An easy answer would be to simply choose a door randomly and ask if it leads to death, then go through one of the others if the answer is "yes". This should guarantee survival, since only one door leads to death, per the warden's rules. It would have a 50% chance of freedom and 50% chance of going through door #3, which probably puts him back in his cell.

• Yes, that's a correct answer, but 50% is not the best possible. (The third door leads indeed back to prison and thus his cell - he somehow had to get into the room, after all.) Commented Mar 10, 2015 at 20:43
• @user66554 He also had to get into the prison, so couldn't he have been brought in by the same door that leads to freedom? Commented Mar 10, 2015 at 21:15
• Well, he came from prison, so bringing him through the door to freedom would be strange. Of course, you can assume so and attempt to solve it by treating the third door as unknown, but then you must account for the possibilities that it might be a second door to freedom, a second door to death or a door back to prison. Commented Mar 10, 2015 at 21:31

I'm going to make 2 assumptions.

1. The warden doesn't lie.

and

1. Because hints are buried in the language, incorrect grammar is essentially a lie. So the grammar of his responses is correct.

Now, I can see 2 interpretations of his comments:

In the first interpretation, in his statement

"One of these doors leads to freedom, the other door [note the singular] leads to death."

the phrase "the other door" imples that only 2 of the objects I see are actually doors. The third is something else (a painting, a false front, an illusion, etc.). I should be able to identify through inspection which one is not a door. At that point I have 2 choices, one Yes/No question, and I'm free.

The second interpretation (which I consider a bit sketchy) is that his use of "the other" is tied to the statement that follows it.

"One of these doors leads to freedom, the other door [note the singular] leads to death. You may choose one door for me to unlock."

He said "One...the other" because only 2 of them are locked, hence you're choosing one of the 2 locked doors, one to freedom, one to death. The third isn't locked and goes into the bathroom, or the prison, or more death, whatever. Doesn't matter. I identify the locked ones, and I'm back to 2 doors, 1 question, and freedom.

• No. Since the mapping of the three doors "left"/"middle"/"right" to the things behind them "freedom"/"death"/"prison" is not known (and the warden didn't mention "prison"), you cannot differentiate the doors from each other in that manner. They are all three indistinguishable and locked. Commented Mar 11, 2015 at 0:15
• If the statement "[note the singular]" is intended to indicate that only one of the doors leads to death, then it should be written, "one other door leads to death", not "the other door [note the singular] leads to death". Your current wording implies there are only 2 doors, which you have indicated is not the case. Commented Mar 11, 2015 at 15:04

My interpretation is thus:

1. There are 3 doors.
2. One leads to freedom and one to death, and the third door leads to the unknown.
3. The prisoner does not need to use his second question to survive.

The problem as posed has been solved several times above and the best solution is 2/3 chance of freedom and 1/3 chance of survival. However, the OP has provided us two additional pieces of information that cannot be deduced purely logically from the posed problem, but clarify the third given statement.

1. (From the OP) Staying in the cell would result in death.
2. (From the OP) The third door returns to the prison, which would constitute survival.

Therefore, the prisoner must be able to leave the cell through one of the two doors that would allow him to survive without asking a second question. Then the prisoner must be able to identify either the door that will kill him or one of the doors that will not kill him. Therefore, one of the doors must be marked, unlocked, transparent, etc. If he can identify any of the three doors unambiguously, he can identify all three, giving a 100% chance of freedom. If he can ambiguously identify one of the doors that will allow him to survive, he can ask "Is going through this door better for me than going through [specific other door]?". If the marked door actually leads to freedom, the answer will always be no, the prisoner will choose the marked door, and he will have a 100% chance of freedom. If the marked door leads back to prison, he will have a 50/50 chance of identifying the door leading to freedom, giving him a 50% chance of freedom and a 100% chance of survival.

• Almost. No door is marked, unlocked, transparent etc. but the information from the OP suffices to select which door to go through to survive without asking a second question. Commented Mar 11, 2015 at 0:37

"Will I survive if I go through the left or the middle door?"

If the Warden answers "Yes", then pick one at random. 50% chance at freedom.

If the Warden answers "No", then go through the other door. 50% chance at freedom.

Your first, most obvious option is to ask about one of the doors specifically. That is, to ask "Does [door A] lead to freedom?" Then, if it does, choose that door, and if not, choose one of the others at random. That first guess has a $1/3$ chance of being correct. If incorrect (a $2/3$ chance), your second guess has a $1/2$ chance of being right, which means there is a $1/3$ chance of this strategy leading to freedom on the second try. So, this gives a $2/3$ total chance of gaining freedom, and a $1/6$ chance of either death or the third option.

As far as I can tell, this is the highest chance you can get of guessing the door to freedom. However, if there is a way to answer the question, "what was behind the third door?" from the information given, then I am overlooking something that could probably give a better answer.

The question asks to minimize the probability of guessing the third door. We can do this and still get a $2/3$ chance of freedom, although this does, of course, increase the chance of death to $1/3$. Personally, I would rather exclude the possibility of death and risk the third door. But, to get rid of the third possibility, you should ask this: "Is the third door (or, the door leading to the third result) to the right of the door to freedom, when viewed from the side of the room with the glass wall?" If the answer is "yes", choosing the leftmost door gives a $2/3$ chance at freedom and $1/3$ chance of death. If the answer is "no", choosing the rightmost door gives the same result.

Asking this question about the "death" door (instead of the third door) allows the prisoner a $2/3$ chance at freedom and no possibility of death, so the warden is correct, you don't need the second question to survive.

• There's a misunderstanding: the prisoner wasted his first question on asking "What's behind the third door?", hence asking "is X the death door?" (which, as y'all noticed, gives 100% survival and 2/3 freedom) already is the second question which, according to the warden, is not necessary. Commented Mar 10, 2015 at 21:16
• @user66554 Wait, so you're saying that the prisoner doesn't need to ask any more questions to guarantee survival? Unless you mean that he can just stay in the cell, that doesn't make sense. Commented Mar 10, 2015 at 21:30
• Yes, survival was not guaranteed before the warden told him that he doesn't need to ask any more questions to guarantee survival. But once the warden said so that was additional information that permits survival without asking another question. Nope, he can't stay in the cell without dying of thirst. Commented Mar 10, 2015 at 21:47

The prisoner just woke up. So, he probably is lying down still. In that case, there is a room on either of his sides and one in the ceiling/roof. The warden singles out the other door and says 'one of these doors' which could mean that he is in the room(behind glass door) at the same level(ground) as the two doors. The prisoner can ask one question to eliminate one of the doors and take the other one.

I know there are some assumptions, but hey, the question is worded that way.

• The door in the ceiling might still have been the one leading to freedom. And no, he isn't lying down. I amended the question to explicitly mention his posture instead of omitting it. Commented Mar 10, 2015 at 21:44

There is three door, lets name it A, B, C doors.

Lets assume there is three state for these doors Death(D), Free(F), Prison(P).

____A___|____B__|____C__|
-----D-----|-----F-----|-----P-----|
-----D-----|-----P-----|-----F-----|
-----F-----|-----D-----|-----P-----|
-----F-----|-----P-----|-----D-----|
-----P-----|-----F-----|-----D-----|
-----P-----|-----D-----|-----F-----|

These are all states there can be.

Then we ask "If A is not the free door then A or B should be death door, right?"

If he says "no" that means, A is free door, or, A is prison door and B is free door.

Prisoner should pick A as a safe choice.

If he says "yes" that means, A or B is death door, and, A is not free door.

In that condition prisoner should pick C as a safe choice.

That question guarantee prisoner's survival and give prisoner %66.7 chances of freedom.

• Your logic is wrong. If A is the free door, he would say "Yes" because the statement is vacuously true. He would only say "No" if A leads to prison, B leads to freedom, and C leads to death. You gain almost no information from that question. Commented Jun 4, 2015 at 16:02
• Thanks, i try to cover it, sorry for rookie(or dummy) mistake i made. Commented Jul 20, 2015 at 8:38

Give me liberty or give me death!

Does this door lead to prison?

I'm certain this isn't the answer you're looking for, but it certainly reduces the probability of going to

prison to 0%

• I gave you the upvote to bring you back to zero. Your answer clearly identifies your intention. Commented Mar 11, 2015 at 19:46
• But it doesn't answer the question. Every answer will clearly identify the user's intention but may be not even close to answering the question itself. Commented Mar 11, 2015 at 21:09

The prisoner turns each knob to determine which of the three doors is unlocked, but does not open the unlocked door. He then asks the warden which of the remaining doors leads to freedom.

Alternatively, the prisoner inspects the doorknobs to find evidence of use (fingerprints, residual warmth, etc.), operating under the assumption that the door to prison is the only one that has been used. He then asks the warden which of the remaining doors leads to freedom.

If I may... Note this is not the same as the other "command" based answers.

My question: Let's call these doors 'yes', 'no', and 'maybe'. If you wanted to say the name of the correct door would your answer to this question be the same as the name of the correct door?"

If door 'yes': his preference is to say 'yes' and that answer is the same as the correct door so he says 'yes'

If door 'maybe': He can only say 'yes' or 'no' so no matter what his answer could not be the same as the correct door. Thus he answers 'no'.

If door 'no': he wants to say 'no' but then his answer would be the same as the name of the correct door, but if he says 'yes' it is no longer correct. He enters a paradox and cannot honestly answer.

As long as you give him time to answer this method should work perfectly.

• When he doesn't know what to say, he answers "No" according to the OP. Commented Apr 26, 2015 at 18:41
• If he says 'no' then he is lying. If he is incapable of lying, he is incapable of answering.
– JCMK
Commented Apr 26, 2015 at 19:11

The warden gives us the most important hint: "Choose one door, you don't even need the question"

So I just say: "I have chosen: Please unlock the door to freedom" Then I check all three doors, the one which is unlocked leads to freedom.

The question isn't needed and can be "What's for dinner?"

I solved the 3 door riddle:

is the left door the right door?

why:

if yes then it is the right go to left door, and no a right door can not be a left door.