Page 5 of 12
Posted: Sun Aug 05, 2007 7:56 pm UTC
sorry then, Ihad missunderstood the problem and had just a assumed that the random sister was more like a coin toss 50% yes 50% no and didnt care what your question was.

Posted: Sun Sep 16, 2007 8:45 am UTC
How about if you asked a question that took a lot of working out. The middle girl would answer instantly, as she can lie or tell the truth, but the liar MUST ensure she is lying, and the truth teller MUST ensure she is telling the truth? So if the princess take a long time in answering marry her. If she takes little time in thinking do not.

Posted: Sun Sep 16, 2007 6:23 pm UTC
Mr Cool wrote:How about if you asked a question that took a lot of working out. The middle girl would answer instantly, as she can lie or tell the truth, but the liar MUST ensure she is lying, and the truth teller MUST ensure she is telling the truth? So if the princess take a long time in answering marry her. If she takes little time in thinking do not.

The actual answer is a lot more satisfying than that. Its in white at the top of page two. So don't read that if you want to work it out yourself, but you can safely read the first page and get some good hints.

[fannishness]Some of the posts are actually by xkcd!![/fannishness]

And, though you don't have to, you might want to visit the intro thread so we have a vague idea of who we're talking to.

Posted: Mon Sep 17, 2007 6:53 pm UTC

Ask them this question: Are you the middle sister?

The oldest sister will answer no, the youngest yes. The middle sister will answer either yes or no. You simply pick the sister with the unique answer.

Edit: I guess this wouldn't be a solution, because you must select one princess and ask the question of only her.

Posted: Mon Sep 17, 2007 7:40 pm UTC
I've been trying to solve this for a while now and I have yet to find a solution. I decided yesterday to read what was the solution and none of the answers in this tread actually work

Posted: Mon Sep 17, 2007 9:11 pm UTC
nufan wrote:I've been trying to solve this for a while now and I have yet to find a solution. I decided yesterday to read what was the solution and none of the answers in this tread actually work

Ummm... what? Have you looked at

http://forums.xkcd.com/viewtopic.php?p=988#988

??

Its the answer, but you have to highlight the text to see it. If you don't think it works, I'd be interested to hear why not.

Also, here's a hint, in case you want to keep trying.

The only questions that wont have a different answer for the youngest and oldest, with the middle answering arbitrarily, are ones which refer to the sisters

Not much of a hint, but it might help you out somewhat.

Posted: Tue Sep 18, 2007 4:56 am UTC
Thx jestingrabbit

### Re: Three princesses

Posted: Fri Oct 12, 2007 12:44 am UTC
"Are you incestuous lesbians?"
Screw marriage, if the sisters are !ugly, it'll be months before you need to worry about them.

### Re: Three princesses

Posted: Fri Oct 12, 2007 3:45 pm UTC
Caveat: I've only read the first page of this thread, so I'm certain to be repeating a solution that has already been rehashed several times.

I reason as follows:

Spoiler:
The answer from either the eldest or youngest daughter is always either "Yes" or "No", and the middle daughter can give either of those answers. Thus there is no way to distinguish whether the daughter of whom I have asked the question is the middle daughter, and I must always marry someone I haven't spoken to. Having decided this, I only need to make sure that my strategy works both when I'm talking to the eldest and when I'm talking to the youngest. In either case, I need to distinguish between the remaining two daughters, so I will point to one of them, ask a question, and depending on whether the answer is "Yes" or "No", marry either the daughter I'm pointing to, or the daughter that I'm neither speaking to nor pointing at.

Here is a question that works: "Is this the younger of your two sisters?" If the answer is "Yes", I marry the daughter I'm pointing at. If the answer is "No", I marry the other one. In all cases, I marry either the eldest or the youngest.

EDIT: Having now read the whole thread, I see I was missing one important point:

(STRATEGY: Pose the question to the least attractive one!)

### Re: Three princesses

Posted: Tue Oct 16, 2007 1:45 am UTC
Spoiler:

The truth-telling sister will say, 'yes'
the lying sister will say, 'no'
and the middle sister will say either

no if two of them say yes, choose the one that says no, as it can only be the youngest sister, and vice versa

### Re: Three princesses

Posted: Tue Oct 16, 2007 2:32 pm UTC
If you had read through, you would have noted that you only get to ask one sister one question. It'd be too easy if you could ask all three.

### Re: Three princesses

Posted: Thu Oct 18, 2007 6:04 pm UTC
I came to the conclusion to look for the sister who gave a unique answer too, but somehow forgot that you only get one question to one girl.

This isn't a guarenteed question, but if you ask "Are you the youngest?" you get a 1/4 chance of correctly identifying the eldest or youngest. (Might be wrong, don't feel like doing probability right now.) Either of those girls will answer "no", guarenteed, and you have a 2/3 chance of picking them. The middle girl will either say "yes" or "no". If she says "yes", you know she's the middle girl. If she says "no"... Well, you messed up, but on the bright side I think she sounds like the more interesting of the three anyways. Consistent truth or lies is dull and boring.

### Re: Three princesses

Posted: Sat Oct 20, 2007 3:18 am UTC
Im sorry if this has come up already, but I read the first page and the last page. This seems a lot more like people comming up with not good answers out of frustration, so I am not going to wade through 5 pages of it. Some may consider my solution cheating, but it isnt really... I also changed the question from yes/no to true/false for simplicities sake
Spoiler:
Ok so some may consider this cheating but here we go. Ask them the question "If we let A be the answer to the statement you sometimes but not always tell the truth, what is A and A?"
This will force the middle sister to say true, while the other two say false. I believe I answer with only one question of one sister.
A A&&A
Truth Teller: F F
Middle (Truth): T T
Middle (Lie): F T The liar evaluates A as a lie and then also lies about A&&A
Liar: T F The liar evaluates A as true, and then lies about A&&A

Im new at HTML, sorry about the formatting

also how do I find harder pure logic problems (Such as this which require basic knowledge of things not logic ie. not probability math questions or something)?

### Re: Three princesses

Posted: Tue Oct 23, 2007 12:14 am UTC

Spoiler:
The truthful sister points to the one who always lies.
The sister capable of lying will point to either the truthful or untruthful sister
The untruthful sister will point to the truthful sister.

Game, set, match. Take the sister that the questioned pointed to.

Edited for grammatical clarity.

### Re: Three princesses

Posted: Tue Oct 23, 2007 4:42 pm UTC
that is similar to the correct answer in a way, but, unfortunately, it's not a yes/no question.

### Re: Three princesses

Posted: Tue Oct 23, 2007 9:33 pm UTC
It doesn't matter, you can point to a sister and ask if she is the one who lies more. If the answer is yes, choose her, otherwise, choose the other one. It's still a nice new idea for this problem. But it is, in a sense, similar to the already stated solution; actually, I think that any difference between all of the sisters can be used to find the right answer in this same manner (of course, presuming you know about said difference).

OK, I've said nothing useful, you can flame me now

### Re: Three princesses

Posted: Thu Oct 25, 2007 1:15 am UTC
Ummm the answer that you ask which one lies the most is incorrect...
lets say you ask the middle sister, or the youngest sister, "who lies the most?".
Either could point to the middle sister or the oldest; to lie they just cant point to the youngest.
to illustrate: if you ask a liar what one plus one equals, they can answer anything aside from 2. or more generally, if the answer to a question is A, anything not A is a lie; the lie is not nessesarily the opposite of A. if I am facing north and you lie about what direction I am facing, NorthEast is a legitimate answer, because I am not.

That kind of reasoning only works when the solution set is bounded to two choices, sorry :-\

im fairly sure my answer works though *see my previous post*

### Re: Three princesses

Posted: Thu Oct 25, 2007 1:22 am UTC
I think the following comic applies to this riddle. Btw, my opinion is that there is no solution, citing all the reasons described above.

I have to add the extra text, it's brilliant: "And the whole setup is just a trap to capture escaping logicians. None of the doors actually lead out."

### Re: Three princesses

Posted: Thu Oct 25, 2007 1:26 am UTC
Gyvulys624 wrote:Btw, my opinion is that there is no solution, citing all the reasons described above.
Except that the posts immediately above yours all contain valid, non-tricky solutions.

### Re: Three princesses

Posted: Fri Nov 23, 2007 10:37 pm UTC
Why can't I ask the King?

### Re: Three princesses

Posted: Fri Nov 23, 2007 10:48 pm UTC

Because you can't, that's why. Even if you could, what yes/no question would you ask?

There is a very simple question which you can ask of precisely one princess that allows you to identify the eldest or youngest. This makes it one of the best puzzles here. If you had to ask the king, it would be a real 169.

### Re: Three princesses

Posted: Fri Nov 23, 2007 10:49 pm UTC
Is this daughter the one that sometimes tells the truth and sometimes lies?

Then you can get either the true one or the lying one every time.

### Re: Three princesses

Posted: Fri Nov 23, 2007 10:58 pm UTC
J Spade wrote:Is this daughter the one that sometimes tells the truth and sometimes lies?

My communication there wasn't as clear as it might be. I should have said something like "there is a yes/no question which, when asked of any one princess, allows you to identify one of the princesses that is eldest or youngest."

### Re: Three princesses

Posted: Thu Nov 29, 2007 4:31 am UTC
Another fairly straightforward solution:

Ask the middle person: "Is the person on your left the middle child, or are you the truth teller, but not both?" (Logical XOR). If 'Yes', hit up the person on the right, if no, the left.

So we know how we can avoid the flip-flopper. A question arises: is it possible to avoid any of the others? The answer seems like it should be no, since if we are unlucky enough to have picked the flip-flopper (middle daughter) we have no useful information about how to avoid anybody else. HOWEVER, if the middle daughter instead of picking randomly, consciously decided beforehand whether she would lie or tell the truth then I bet such a solution (although meta) would exist.

### Re: Three princesses

Posted: Thu Nov 29, 2007 5:49 am UTC
Do you mean that you want to ask a single question to determine a not-eldest or not-youngest or do you mean that you want to ask a single question to determine the youngest or eldest? The first is probably doable, but second isn't.

### Re: Three princesses

Posted: Fri Nov 30, 2007 2:37 am UTC
I meant the first, but like I said we need a middle child who isn't random in her answer but instead random in her approach to the answer (lying or truthful frame of mind).

The second probably isn't doable, but probably doable in two questions. Its interesting, but I've never seen a good proof that some problem like this is impossible. (except for very trivial cases). For example, before the solution to this one came around, many people thought it would be impossible, but I don't think anybody knew how to try to approach proving it. (Though this is kind of a bad example, since any proof of its impossibility would have obviously been incorrect).

### Re: Three princesses

Posted: Fri Nov 30, 2007 3:00 am UTC
imatrendytotebag wrote:The second probably isn't doable, but probably doable in two questions. Its interesting, but I've never seen a good proof that some problem like this is impossible.

Hm? There are three daughters. You want to find out which of them is the youngest, say. There are three possible solutions, but you only get a single yes or no answer to a single question. There aren't enough bits of information to encode which daughter is the youngest (or middle, or oldest), so clearly it's impossible.

### Re: Three princesses

Posted: Fri Nov 30, 2007 3:30 am UTC
Buttons wrote:
imatrendytotebag wrote:The second probably isn't doable, but probably doable in two questions. Its interesting, but I've never seen a good proof that some problem like this is impossible.

Hm? There are three daughters. You want to find out which of them is the youngest, say. There are three possible solutions, but you only get a single yes or no answer to a single question. There aren't enough bits of information to encode which daughter is the youngest (or middle, or oldest), so clearly it's impossible.

Yeah, of course. Good call.

When I was talking about "some problem like this" I was actually referring to more complicated things (ie variations on the 4 gods problem), but at any rate well said about the impossibility of this one.

### Re: Three princesses

Posted: Sat Dec 01, 2007 3:33 pm UTC
GUYS!!!!!!!!!!!!!

I went to sleep yesterday trying to figure it out, and I have it now.

SPOILER ALLERT!!!!!!!!!
Spoiler:
I would ask the question in one sentence, but it is two hard to understand it unless I break it up.

This is the question I would ask:

Q#1, Is the Random sister lying this turn?

Q#2, If I asked the less truthful of the other two sisters Q#3 about you (pointing at you), would she always be allowed to answer?
Q#3, Is the answer to this question the same as the answer to the question "is that sister (pointing) the sister who sometimes lies and sometimes tells the truth?

I am going to post the explanation on my next post, because it could use an explanation.

### Re: Three princesses

Posted: Sat Dec 01, 2007 3:57 pm UTC
That would help, as would a table with possibilities and consequences.

### Re: Three princesses

Posted: Sat Dec 01, 2007 4:10 pm UTC
This is my explanation for my solution

Spoiler:
The Question:
Q#1, Is the Random sister lying this turn(or now)?

Q#2, If I asked the less truthful of the other two sisters Q#3 about you (pointing at you), would she always be allowed to answer?

Q#3, Is the answer to this question the same as the answer to the question "is that sister (pointing) the sister who sometimes lies and sometimes tells the truth?"

What is the truthful answer to Q#1?
"Q#1, Is the Random sister lying this turn(or now)?"
If I am not talking to the Random sister, then she is not lying because she is not talking.

Truth telling sister: No
Lying sister: No (I asked her a question about this question, so at the end, she will lie regarding the overall question, not each individual question)
Random sister: Yes if she lies in the overall question, no if she doesn't

Now the overall question:
is
(The A to Q#1) not= (the A to Q#2)?

Q#2 is built so that the true answer, when I'm asking each sister is:
(Truth telling sister)= Yes
(Lying sister)= No
(Random sister)= No
I will go into the explanation of Q#2 later.

Truth telling sister:
(The A to Q#1) = No
(the A to Q#2) = Yes
They are different, and she won't lie about it, so she would say {YES}

Lying sister:
(The A to Q#1) = No
(the A to Q#2) = No
They are not different, but she would lie about it, so she would say {YES}

Random sister:
(The A to Q#1) = Yes if she lies in the overall question, no if she doesn't
(the A to Q#2) = No
She has two choice, lie, or not to lie.
Lie (The A to Q#1) = Yes , The two answers are different, but she would say that they are not {No}
Truth (The A to Q#1) = No, The two answers are not different, and she would that they are not different {No}

Lier: Yes
Honest sister: Yes
Random sister: No

I am going to explain the real answer to question #2 in my next post.

Continuing:

Spoiler:
The Question:
Q#1, Is the Random sister lying this turn(or now as I'm asking this question)?

Q#2, If I asked the less truthful of the other two sisters Q#3 about you (pointing at you), would she always be allowed to answer?

Q#3, Is the answer to this question the same as the answer to the question "is that sister (pointing) the sister who sometimes lies and sometimes tells the truth?"

Now, Q#2 is built so that the true answer, when I'm asking each sister is:
(Truth telling sister)= Yes
(Lying sister)= No
(Random sister)= No

Remember, whatever sister I ask, if she lies, she can only lie to the answer to the overall question, not this individual question.

To understand this explanation, you need three objects in front of you, because the RAM of your brain doesn't need this info clogging it up.
Get a "good" object to symbolize the truth telling sister,
a "bad" object to symbolize the lying sister,
and a random or ever-changing object to symbolize the Random sister.

Again, remember, whatever sister I ask, if she lies, she can only lie to the answer to the overall question, not this individual question. The lying will be taken into acount later.

Q#3 is an odd one. If you answer yes, or no, you imply that the sister being pointed at (the sister that the overall question is directed at) is the random sister.
If that sister is the Random sister, you could only tell the truth, by saying Yes or no, both of which imply that the sister being pointed at is the random sister.
If the sister being pointed at is the random sister, only the truth telling sister could answer the question, because the always-lying sister could not imply that the random sister is herself, because that is the truth, and the lying sister must lie.

Remember the "always" in question #2.

Here is an example:

Is the answer to this question the same as the answer to the question "Would you go out with me"?

If you say yes, then yes = the answer to the question "Would you go out with me?"
If you say no, then no does not = the answer to the question "Would you go out with me?",
meaning that the answer to the question "Would you go out with me?" = yes.

If I ask someone this question, no matter what they say, they are implying that they want to go out with me.
If they say "yes" or "no", then they are implying that they want to go out with me.
No this doesn't work on most girls, because, on not being able to answer the question with words, they usually answer with a slap to the face. My face.

Now ask the Truth sister the Q#2
Q#2, If I asked the less truthful of the other two sisters Q#3 about you (pointing at you), would she always be allowed to answer?

Q#3, Is the answer to this question the same as the answer to the question "is that sister (pointing) the sister who sometimes lies and sometimes tells the truth?"

When talking to the truth telling sister, Q#3 would be directed at the always-lying sister, because she is less truthful than the random sister, who occasionally tells the truth.
Could the lying sister always answer the question #3?
Question number three implies that the truth-telling sister is the random sister, which is not true.
Would the lying sister imply something that is not true?
Yes.

(The A to Q#2) when asking these specific sisters:
Truth telling sister= Yes

The less truthful of the other two sisters is still the lying sister.
Could the lying sister always imply that the Random sister is the Random sister?
No, she could, would never.

(The A to Q#2) when asking these specific sisters:
Truth telling sister= Yes
Random sister = No

The less truthful of the other two sisters is the Random sister.
Could the Random sister always imply that the lying sister is the random sister.
If the random sister is lying at the time, yes.
If not, no.
So the random sister could not always answer Q#3, which implies that the lying sister is the random sister.

(The A to Q#2) when asking these specific sisters:
Truth telling sister= Yes
Random sister = No
Lying sister = No

A compilation of all tables:
Spoiler:
The overall question:
is
(The A to Q#1) not= (the A to Q#2)?

Truth telling sister: No
Random sister: Yes if she lies in the overall question, no if she doesn't
Lying sister: No

Truth telling sister= Yes
Random sister = No
Lying sister = No

Truth telling sister:
(The A to Q#1) = No
(the A to Q#2) = Yes
They are different, and she won't lie about it, so she would say {YES}

Lying sister:
(The A to Q#1) = No
(the A to Q#2) = No
They are not different, but she would lie about it, so she would say {YES}

Random sister:
(The A to Q#1) = Yes if she lies in the overall question, no if she doesn't
(the A to Q#2) = No
She has two choice, lie, or not to lie.
Lie (The A to Q#1) = Yes , The two answers are different, but she would say that they are not {No}
Truth (The A to Q#1) = No, The two answers are not different, and she would that they are not different {No}

Lier: Yes
Honest sister: Yes
Random sister: No

I just noticed Aaronspook's answer from over a year ago.
Ok guys, my explanation could certainly use some fixing up.

I am going to wait until I get enough feedback about it before I merge all of the explanations and put them together.

Thanks!

thanks jr

### Re: Three princesses

Posted: Tue Jun 03, 2008 7:32 pm UTC
Swamp, I am unsure if I agree with your answer. It seemed tobe three questions asked in one?

A much shorter or easier question would be:
Spoiler:
You ask Person C, "Is person B older than Person A?" and you marry whichever one is younger.

Table would be:
Spoiler:
If person C was the liar she would answer that the Oldest(truth teller) was younger than the middle aged girl.
If person C was the truth teller, the younger person would be the Liar and you can marry her.
If person C was the middle girl, doesn't matter who you marry if you don't marry person C. So whoever she said is younger it is either the truth teller or the liar. And yay you have a predictable fiance.

Now maybe your's is saying the same thing, But I just didn't understand the answer haha.

### Re: Three princesses

Posted: Wed Jun 04, 2008 9:10 am UTC
Here's another version of the problem that I've solved. (actually, the partial solution to it is floating in this thread somewhere.)

Same setup with the princesses and your preference, different procedure. This time the king brings to you one princess randomly. You may ask one yes-no question to her. After she responds, you must choose to accept or reject her. If you reject her, however, the king chooses you bride randomly from the remaining two.

### Re: Three princesses

Posted: Wed Jun 04, 2008 5:14 pm UTC
My question works for both versions of the question.

The Random sister would answer no, so I would ask the king/father to marry one of the others.
Both the lying sister and the truthful sister would answer yes, so I would marry the sister in front of me.

My question in two versions:

Spoiler:
Q#1, Is the Random sister lying this turn(or now)?

Q#2, If I asked the less truthful of the other two sisters Q#3 about you (pointing at you), would she always be allowed to answer?

Q#3, Is the answer to this question the same as the answer to the question "is that sister (pointing) the sister who sometimes lies and sometimes tells the truth?"

The Good Version:

Spoiler:
Is the real answer to "Is the Random sister lying this now?" different from the real answer to "Is the less truthful of your two sisters always able to imply that you are the princess that sometimes lies and sometimes tells the truth?"?

TimM1104 wrote:Swamp, I am unsure if I agree with your answer. It seemed tobe three questions asked in one?

No, the actual question is the second version, which is obviously one question.

### Re: Three princesses

Posted: Fri Jun 06, 2008 12:08 pm UTC
I feel i may have discovered an alternate solution, assuming that "i dont know" is a legitimate answer"

if i have stubled across the lying sister, she will likely say yes, or no, and thus be recognisable as the middle sister.

either of the other sisters will be unable to predict their sisters behavior, and thus answer "i dont know"

This probably isnt a legit answer, but i like it anyway, even if its not as foolproof as the other way.

### Re: Three princesses

Posted: Fri Jun 06, 2008 7:57 pm UTC
jestingrabbit wrote:Do you mean that you want to ask a single question to determine a not-eldest or not-youngest or do you mean that you want to ask a single question to determine the youngest or eldest? The first is probably doable, but second isn't.

isn't this trivial? Just ask a question whose value you already know. E.g., "1+1=2?" The oldest will answer no, the middle random, the youngest yes. If the princess you ask answers yes, then you know that princess is not the youngest. Vice versa for non-oldest.

### Re: Three princesses

Posted: Sat Jun 07, 2008 5:33 am UTC
thc wrote:
jestingrabbit wrote:Do you mean that you want to ask a single question to determine a not-eldest or not-youngest or do you mean that you want to ask a single question to determine the youngest or eldest? The first is probably doable, but second isn't.

isn't this trivial? Just ask a question whose value you already know. E.g., "1+1=2?" The oldest will answer no, the middle random, the youngest yes. If the princess you ask answers yes, then you know that princess is not the youngest. Vice versa for non-oldest.

But that's asking three questions, one to each of the daughters. The trick is to identify a non-random-answerer with only one question to one daughter.

### Re: Three princesses

Posted: Sat Jun 07, 2008 7:32 pm UTC
jestingrabbit wrote:
thc wrote:
jestingrabbit wrote:Do you mean that you want to ask a single question to determine a not-eldest or not-youngest or do you mean that you want to ask a single question to determine the youngest or eldest? The first is probably doable, but second isn't.

isn't this trivial? Just ask a question whose value you already know. E.g., "1+1=2?" The oldest will answer no, the middle random, the youngest yes. If the princess you ask answers yes, then you know that princess is not the youngest. Vice versa for non-oldest.

But that's asking three questions, one to each of the daughters. The trick is to identify a non-random-answerer with only one question to one daughter.

No, what I mean: ask "1+1=2," if you get a "no" answer, then you know that that daughter is not the oldest. If you get a "yes" answer, that daughter is not the youngest. So no matter what response you get, you determine non-eldest or non-youngest.

### Re: Three princesses

Posted: Sun Jun 08, 2008 2:14 am UTC
thc wrote:
jestingrabbit wrote:
thc wrote:
jestingrabbit wrote:Do you mean that you want to ask a single question to determine a not-eldest or not-youngest or do you mean that you want to ask a single question to determine the youngest or eldest? The first is probably doable, but second isn't.

isn't this trivial? Just ask a question whose value you already know. E.g., "1+1=2?" The oldest will answer no, the middle random, the youngest yes. If the princess you ask answers yes, then you know that princess is not the youngest. Vice versa for non-oldest.

But that's asking three questions, one to each of the daughters. The trick is to identify a non-random-answerer with only one question to one daughter.

No, what I mean: ask "1+1=2," if you get a "no" answer, then you know that that daughter is not the oldest. If you get a "yes" answer, that daughter is not the youngest. So no matter what response you get, you determine non-eldest or non-youngest.

But you don't get to pick which kind of person you're identifying using that strategy. After you ask the question you either determine a non-eldest or a non-youngest, but if you set out to determine a non-eldest and you get a truthful response you don't know whether you are dealing with the truthful or the random, and so haven't IDed a non-eldest person.

But you're right that you can determine either a non-eldest or a non-youngest with one question, so long as you don't care which it is that you want to identify.

### Re: Three princesses

Posted: Mon Jun 09, 2008 8:57 am UTC
Without asking any question to the girls, a priori you have a 2/3 chance to make a good decision.

Suppose that you were unable to come up with the right question to ask, and instead just ask one sister "Are you the oldest one?" and she answers with yes or no. Does this give you any new information?