## Three princesses

A forum for good logic/math puzzles.

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### Re: Three princesses

Actually, it gives you a 5/6 which is even better?

Eldest or Youngest is asked - 2/3 chance

If the middle is asked, and she has a 50/50 chance of saying yes/no, it means there is a 1/6 chance of you asking the middle sister AND she says 'yes' (1/3*1/2 =1/6)

So 2/3+1/6 (situations in which you wouldn't marry the middle one) = 5/6! Woo!

However, given she's mischievous (as mentioned in the OP), she would probably say 'no' in order to screw you over (perhaps you're dashingly handsome and she really wants you to pick her?). Thus putting your chances at 2/3 of not marrying the middle sister, the same as guessing randomly.

There is an answer that's been stated before, obviously, and it gives you 100% certainty of not marrying her.
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tjwoods
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### Re: Three princesses

Actually, I think there is an easier solution that doesn't require asking a question to any of the princesses. If the problem is "What yes or no question can you ask which will ensure you do not marry the middle sister?", then the solution I see is to ask yourself "Do I really want to need to devise clever meta-questions and solve logic problems every time I interact with my in-laws?" Then you answer "#\$^ No!", walk away and do not ever have to worry about marrying the middle sister.

fjafjan
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### Re: Three princesses

jestingrabbit wrote:
rddrgn wrote:(skipped some pages so dont know if its already been answered)
Spoiler:
"will the middle sister answer truthfully (or not truthfully, it doesn't matter) if i were to ask her a question?"

the truthful sister and the lying sister cannot give an answer because they cannot predict what the middle sister would say
the middle sister can say either yes or no. if you get an answer its the middle sister

It has been answered, and without resorting to questions which are sometimes unanswerable.

Yeah I asked this questino to some friends, and the next day one came at told me this answer.
I personally think it's not valid as it's technically not a yes or no question. Or at least I would argue that it's not a yes or no question, since the answer will not be yes or no.
//Yepp, THE fjafjan (who's THE fjafjan?)
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CortoPasta
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### Re: Three princesses

You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry. The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time.

As you will be forever married to one of the princesses, you want to marry the eldest (truth-teller) or the youngest (liar) because at least you know where you stand with them.

The problem is that you cannot tell which sister is which just by their appearance, and the King will only grant you ONE yes or no question which you may only address to ONE of the sisters. What yes or no question can you ask which will ensure you do not marry the middle sister?

I've only read a little bit from the past nine pages, but I didn't see anything about being able to point at a sister/have a third answer(i.e. which sister doesn't respond/etc., so I'm not sure why we consider those answers

It seems like a simple XOR logic gate, but verbally expressing it is tricky.

Spoiler:
Are you one of the youngest two daughters, or would you say that you are the middle daughter?

Oldest(Truth): No
//Not one of the youngest two daughters, nor would she say she is the middle daughter
Middle(While telling the truth): Yes
//Is one of the youngest two daughters, and would say she is the middle daughter
Middle(While telling a lie): Yes
/*Would say no to being one of the youngest two daughters, but the question "or would you say that you are the middle daughter?" Would be a double negative. {i.e. Because her answer would be no to "Are you the middle daughter?", she would lie and say "yes"{i.e. Because her answer(as a liar) would be no to "Are you the middle daughter?", she would lie and say "yes" to the part "would you say that you are the middle daughter?"}*/
Youngest(Liar): No
/*Would say no to being one of the youngest two daughters, but would have the opposite response to the double negative second part of the question and respond no {i.e. Because her answer(as a liar) would be yes to "Are you the middle daughter?", she would lie and say "no" to the part "would you say that you are the middle daughter?"} */

So using the answer in the spoiler box, pick the sister that says no.

To bad this is just make believe, cause in the real world they would all just flip-flop all the time :p

Edited: Just rearranged spoiler to make it easier to read

douglasm
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### Re: Three princesses

Not bad, CortoPasta. Now try the interpretation of the middle sister where she ignores your question and mentally flips a coin to determine her answer.

ice__101
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### Re: Three princesses

Has this been answered yet? I think I have the answer, skimmed through, but didn't see it. Let me know if this will work.

Spoiler:
Ask All Three if they are the middle sister
The oldest one will say No
The Youngest one will say Yes
You don't know what the middle sister will say yet.

If you get two Yes's and 1 No, go with the girl who said no.
If you get two No's and 1 Yes, go with the girl who said yes.

You wont know if you got the younger or older daughter, but you will know you got at least one of them and can figure out which one easily enough at a later date.

Moonbeam
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### Re: Three princesses

ice__101 wrote:Has this been answered yet? I think I have the answer, skimmed through, but didn't see it. Let me know if this will work.

Spoiler:
Ask All Three if they are the middle sister
The oldest one will say No
The Youngest one will say Yes
You don't know what the middle sister will say yet.

If you get two Yes's and 1 No, go with the girl who said no.
If you get two No's and 1 Yes, go with the girl who said yes.

You wont know if you got the younger or older daughter, but you will know you got at least one of them and can figure out which one easily enough at a later date.

Fleshpiston wrote:Greetings all, here is a logic puzzle I read online a while back.

You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry. The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time.

As you will be forever married to one of the princesses, you want to marry the eldest (truth-teller) or the youngest (liar) because at least you know where you stand with them.

The problem is that you cannot tell which sister is which just by their appearance, and the King will only grant you ONE yes or no question which you may only address to ONE of the sisters. What yes or no question can you ask which will ensure you do not marry the middle sister?

i.e. you're allowed to ask one question to only one of the sisters (obviously you won't know which one you're asking). You're asking one question to all 3 sisters .........

CortoPasta
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### Re: Three princesses

douglasm wrote:Not bad, CortoPasta. Now try the interpretation of the middle sister where she ignores your question and mentally flips a coin to determine her answer.

One miraculous answer per day. Maybe tomorrow...

Loafer
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### Re: Three princesses

The middle girl snags things up, true.
To solve this, remember my clue:
When querying a princess,
phrase it so that the lass
you choose is from the other two.

TheChewanater
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### Re: Three princesses

If the answer is no, run screaming into the night.

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dedalus
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### Re: Three princesses

Even better, try it in the case where the middle sister is trying to get you to marry her.
doogly wrote:Oh yea, obviously they wouldn't know Griffiths from Sakurai if I were throwing them at them.

cougaman
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### Re: Three princesses

couldn't you just ask the king..which is your middle daughter? Then marry the other?

saudifreestyler
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### Re: Three princesses

well i think ive got an answer well a question at least
Spoiler:
ok, if the middle princess has a 50-50 chance of telling the truth or a lie then logically she would tell a lie followed directly by the truth, so if if you ask her a simple question ive just thought up then she should be taken out from the other princesses.

if you asked it to the truthful princess then her answer to it would be no, are you a princess yes, is that a lie no
if you asked it to the untruthful princes then her answer to it would be no, are you a princess no, is that a lie no
so both the truthful and untruthful princess answer no
but if you ask the other princess her answer would be yes always
if she lied first- are you a princess no, is that a lie yes
if she was truthful first then are you a princess yes, is that a lie yes

so by this logic if she answers yes then she is the 50-50 princess but if she answers no then she is the truthful or untruthful princess the one you want

Goldstein
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### Re: Three princesses

saudifreestyler wrote:if the middle princess has a 50-50 chance of telling the truth or a lie then logically she would tell a lie followed directly by the truth

In the same way that, logically, a coin that has been flipped and came up heads will come up tails next time?
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dedalus
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### Re: Three princesses

Seriously guys; the puzzle can be solved with the condition that the middle sister will answer true or false with the intent of marrying you if at all possible... and this in itself is a hint as to the answer.
doogly wrote:Oh yea, obviously they wouldn't know Griffiths from Sakurai if I were throwing them at them.

Lexodus
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### Re: Three princesses

Which one's hotter?

MSTK
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### Re: Three princesses

I remember this topic from a long time ago, and I just stopped by to say that this puzzle and its answer are among the most elegant I have ever seen. It doesn't involve any trickery, unanswerable questions, lateral thinking (albeit one major leap)....it just asks for pure, elegant logic.

Note that this can be solved with a meta-question, but why bother? The true answer is so much simpler and doesn't even have to boggle the mind.

Xami
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### Re: Three princesses

I have not read through all the pages but it seems an answer is still being sought?
Spoiler:
I put this on a piece of paper

Good(Truth) Good(Liar)

I began with connecting lines that if asked to the eldest would point me to the youngest and settled on this
Of the other two - who is younger?
The eldest would point to the young liar.

Applying this to the young liar - she points to the eldest
Applying this to the middle results in/.. oh who cares she will not pick herself.
Just pick whoever the girl points to?

You won't know immediately if you found the liar or not, but I'm sure you can figure it out soon - you are allowed to converse with her freely enough after marriage ya?
Edit: It would seem this was answered on page 2 but wordier?

skeptical scientist
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### Re: Three princesses

Xami wrote:I have not read through all the pages but it seems an answer is still being sought?
Edit: It would seem this was answered on page 2 but wordier?

Indeed. Answered a long time ago, it's just that people seem to keep inventing new solutions based on bizarrely worded questions and even more bizarre interpretations of how the middle sister would be forced to answer them.

If I were a mod, I would edit the first post to remove all solutions besides the intended one (or other solutions based on the same idea), and then add a footnote explaining that there is a very satisfying solution to this puzzle that doesn't require rules-lawyering the concept of truth.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

zaratustra
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### Re: Three princesses

You've been offered the hand of three princesses: One always tells the truth, one always lies, and one says lies and truths at random. How can you make sure to marry the one with the biggest breasts?

TheChewanater
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### Re: Three princesses

Oh oh oh! I got it! I think...

It kind of depends on what you mean by "sometimes lies and sometimes tells the truth". For this to work you must assume that she even lies to herself.

Spoiler:
Have them stand in a triangle, like this. Make them face outwards, so 2 is to 1's right, 3 is to 2's right, and 1 is to 3's right.
..1..
3...2

If she says No, marry 1.
If she says Yes, marry 3.

PROOF:
If 1 is the truther, she'd say no either way, and you'd marry 1 (the truther).

If 1 is the middle, she'd say Yes either way, and you'd marry 3, which could be either.

If 1 is the lier, she'd say No if 2 was the truther, and you'd marry 1 (the lier), and she'd say Yes if 2 was the middle, and you'd marry 3 (the truther).
Last edited by TheChewanater on Sat Oct 03, 2009 3:25 am UTC, edited 2 times in total.

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No one can agree how to count how many types of people there are. You could ask two people and get 10 different answers.

Xami
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### Re: Three princesses

Spoiler:
If 1 is the middle, she'd say Yes either way, and you'd marry 3, which could be either.

Or she could lie/tell truth and say no...

skeptical scientist
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### Re: Three princesses

Congratulations, you have come up with a completely meaningless question. The question asks one to make a comparison involving one's own answer to the question. In other words, to figure out what the question is asking, you must already know the answer.

When attempting to provide a solution to this puzzle, ask yourself the following question: "For my solution to work, do I need to know how the youngest princess would answer if she were asked what her sister would say if she asked her whether her principle difficulty in answering this question has more to do with its paradoxical nature or the large number of ambiguous pronouns?" If the answer is yes, then you're doing it wrong.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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TheChewanater
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### Re: Three princesses

skeptical scientist wrote:

Congratulations, you have come up with a completely meaningless question. The question asks one to make a comparison involving one's own answer to the question. In other words, to figure out what the question is asking, you must already know the answer.

When attempting to provide a solution to this puzzle, ask yourself the following question: "For my solution to work, do I need to know how the youngest princess would answer if she were asked what her sister would say if she asked her whether her principle difficulty in answering this question has more to do with its paradoxical nature or the large number of ambiguous pronouns?" If the answer is yes, then you're doing it wrong.

Um... it seems to work. The wording was a little messed up since I had another wrong answer, which I edited before posting, and I didn't erase the whole thing.

Could you explain why this isn't right? It doesn't let you determine who is who, but it lets you know one of them that is either 1 or 3. I think.

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No one can agree how to count how many types of people there are. You could ask two people and get 10 different answers.

Xami
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### Re: Three princesses

You're missing the point of the middle sister.
If 1 is the middle, she'd say Yes either way, and you'd marry 3, which could be either.

Her whole reason for being is she will sometimes tell the truth and vice versa. You're flaw stems from deciding she will lie to herself.

Tip
Spoiler:
You can 'win' this game in 3 words

skeptical scientist
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### Re: Three princesses

As I said: it is impossibly to answer the question truthfully, i.e. by evaluating the question, deducing a correct answer, and saying it. In order to know how to answer it, you need to already know your answer, so you have a circular paradox preventing being able to answer it.

Now there's the issue of the question making a possibly false assumption. "The way you always answer" is not well-defined when the question is posed to the middle sister. And there are other issues as well.

Anyways, correct solutions to this puzzle are not very convoluted ways of asking questions. A correct solution consists of a simple idea, together with a simple question, and is obviously correct once you find it.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

Anarra
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### Re: Three princesses

Spoiler:
You could ask them all the same question and go for the only unique answer, you then will no that you have either the lier or the truthteller and can find out which one later on.

If I'm not mistaken the question is asking you to get rid of the middle sister not to pick one of the other 2.

TheChewanater
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### Re: Three princesses

the king will only grant you ONE yes or no question which you may only address to ONE of the sisters

http://internetometer.com/give/4279
No one can agree how to count how many types of people there are. You could ask two people and get 10 different answers.

Xami
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### Re: Three princesses

What I don't understand is how intuitive this and the 'Blue Eyes' solutions are to me, but there are other much more difficult problems (to me) on this site that everyone just seems to be like - duh, of course! That was easy!

jestingrabbit
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### Re: Three princesses

Xami wrote:What I don't understand is how intuitive this and the 'Blue Eyes' solutions are to me, but there are other much more difficult problems (to me) on this site that everyone just seems to be like - duh, of course! That was easy!

I think the neatness of these puzzles is what leads to this. There aren't any confounding bits of info, nothing is spare or redundant. There is also, I think especially with this one, such an absolute elegance to the solution.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Jibun Wo
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### Re: Three princesses

I have done this before,
Spoiler:
you ask the one what the other would say and since if it is the truther it would say the other would say the wrong, and if it is lie person, she would say the lie of the times, I asked what the other would say and my mom got scared and said you movin with yall auntie an uncle in bel air. I whistled for a cab and when it came newar the license plate said "fresh" and there be dice in the mirror, if anything I could say this cab was rare, but i said na forget it, Yo holmes to bel-air!

jestingrabbit
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### Re: Three princesses

The problem with that is that there are three princesses, one of which is random. So not just a truth teller and a liar to deal with.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

EireannX
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### Re: Three princesses

Spoiler:
Ask Sister Alpha if Sister Beta lies more than sister Delta.

If the answer is Yes, Marry Beta. If No, Marry Delta

The trick is that if you get miss indeterminate answering your question, then both Beta and Delta are good choices.

jestingrabbit
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### Re: Three princesses

EireannX wrote:
Spoiler:
Ask Sister Alpha if Sister Beta lies more than sister Delta.

If the answer is Yes, Marry Beta. If No, Marry Delta

The trick is that if you get miss indeterminate answering your question, then both Beta and Delta are good choices.

That's a pretty good one, and is essentially it,
Spoiler:
but there is a way to determine which sister lies more in a manner that is definite. atm, you could imagine a situation where a sister might be unsure as to how to answer, in that the middle sister is completely arbitrary in her answers, so she might tell the truth all the time just to spite us.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

tekk
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### Re: Three princesses

here's my stab at it(this assumes that you ask ONE question to all three at the same time, you were kind of ambiguous on that)
Spoiler:
ask ANY question, so let's ask do you have feet, assuming you have no amputees, there will be either a yes yes no or a no no yes, you just marry the one who says no in the former case and yes in the latter

TheChewanater
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### Re: Three princesses

tekk wrote:here's my stab at it(this assumes that you ask ONE question to all three at the same time, you were kind of ambiguous on that)
Spoiler:
ask ANY question, so let's ask do you have feet, assuming you have no amputees, there will be either a yes yes no or a no no yes, you just marry the one who says no in the former case and yes in the latter

the King will only grant you ONE yes or no question which you may only address to ONE of the sisters

http://internetometer.com/give/4279
No one can agree how to count how many types of people there are. You could ask two people and get 10 different answers.

tekk
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### Re: Three princesses

TheChewanater wrote:
tekk wrote:here's my stab at it(this assumes that you ask ONE question to all three at the same time, you were kind of ambiguous on that)
Spoiler:
ask ANY question, so let's ask do you have feet, assuming you have no amputees, there will be either a yes yes no or a no no yes, you just marry the one who says no in the former case and yes in the latter

the King will only grant you ONE yes or no question which you may only address to ONE of the sisters

whoops you could get the same result from asking them all htat one question

TheChewanater
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### Re: Three princesses

tekk wrote:
TheChewanater wrote:
tekk wrote:here's my stab at it(this assumes that you ask ONE question to all three at the same time, you were kind of ambiguous on that)
Spoiler:
ask ANY question, so let's ask do you have feet, assuming you have no amputees, there will be either a yes yes no or a no no yes, you just marry the one who says no in the former case and yes in the latter

the King will only grant you ONE yes or no question which you may only address to ONE of the sisters

whoops you could get the same result from asking them all that one question

the King will only grant you ONE yes or no question which you may only address to ONE of the sisters

http://internetometer.com/give/4279
No one can agree how to count how many types of people there are. You could ask two people and get 10 different answers.

Quylyn
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### Re: Three princesses

Not sure if this answer has been given yet or not, but it works:

Spoiler:
Ask sister A, "Is B younger than C?"

If sister A is the eldest, she will answer truthfully. Marry the one A says is younger -- it will be the youngest sister.
If sister A is the youngest, she will lie. Marry the one A says is younger -- it will be the eldest sister.
If sister A is the middle sister, it doesn't matter if she lies or tells the truth. Marry the one A says is younger -- it will be either the youngest or eldest sister.

So no matter which sister A is, marry the one that she says is younger.

jestingrabbit
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### Re: Three princesses

Well done Quylyn. That is the answer that I consider best.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.