## Three princesses

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jestingrabbit
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### Re: Three princesses

AzraelFish wrote:Alright...having read all six pages, and realizing several solutions have been given, I'm not even going to bother--suffice it to say, I thought about it enough for my own satisfaction, and came up with, if not a workable solution, something at least close enough to make me happy with it. So, what I DO want to say...rather than finding a question that would tell you that yes, you know that one in particular is not the middle sister, I'd rather figure out exactly whom the middle one is--if I were going to marry someone, I'd like it to be her. It's no fun talking to someone, day in and day out, if it's always...predictable. I, at least, would want some surprise from my wife.
I may post again in a couple days, if I figure out how to say for certain which is the middle sister. =P
Until then...peace, yo's!

--Fish

This is impossible. You need more than one bit of information to work out who the middle sister is, but you only get one bit from the question. Two questions would give it to you pretty easily though (think about it).
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

DrakeSD
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### Re: Three princesses

I haven't read all seven pages, so I don't know if someone has come up with this yet, but there is only one way I can thing of to do this, and it only works if the middle sister can say "I don't know"

Spoiler:
You ask "If I asked you the question 'do you ever lie' would you say no."
The eldest would say 'yes' as her answer to 'do you ever lie' would be 'no.'
The youngest would say 'no' as her answer to 'do you ever lie' would also be 'no,' but she must lie about it.
The middle sister must say 'I don't know,' because her answer could go either way, depending if she was telling the truth or not. If she was telling the truth, her answer to 'do you ever lie' would be yes. If she was lying, it would be 'no.'
Owimacattsor!

jestingrabbit
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### Re: Three princesses

DrakeSD wrote:I haven't read all seven pages, so I don't know if someone has come up with this yet, but there is only one way I can thing of to do this, and it only works if the middle sister can say "I don't know"

Spoiler:
You ask "If I asked you the question 'do you ever lie' would you say no."
The eldest would say 'yes' as her answer to 'do you ever lie' would be 'no.'
The youngest would say 'no' as her answer to 'do you ever lie' would also be 'no,' but she must lie about it.
The middle sister must say 'I don't know,' because her answer could go either way, depending if she was telling the truth or not. If she was telling the truth, her answer to 'do you ever lie' would be yes. If she was lying, it would be 'no.'

There's a much better solution. HInt in the spoiler.

Spoiler:
think about questions which refer to the other sisters.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

George2
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### Re: Three princesses

this puzzle is very similar to that one, but easier.

jestingrabbit
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### Re: Three princesses

George2 wrote:this puzzle is very similar to that one, but easier.

We did that one a while back.

viewtopic.php?f=3&t=273
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Crosby
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### Re: Three princesses

(Please pardon me if this was already said.)

A simple solution:
Spoiler:
Ask "Which one of your sisters lies more (than sometimes)?"
Then pick the one she points to.

Let me explain:
If we eliminate as a candidate the sister we pose the question to, then, if we ask:
truth-teller - she points to the liar (success)
flipper - she points to either the liar or the truth-teller (success)
liar - she points the truth-teller (success)

jestingrabbit
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### Re: Three princesses

Crosby wrote:(Please pardon me if this was already said.)

A simple solution:
Spoiler:
Ask "Which one of your sisters lies more (than sometimes)?"
Then pick the one she points to.

Let me explain:
If we eliminate as a candidate the sister we pose the question to, then, if we ask:
truth-teller - she points to the liar (success)
flipper - she points to either the liar or the truth-teller (success)
liar - she points the truth-teller (success)

that's pretty much the optimal answer, but

Spoiler:
being less truthful is the same as being younger.

and spoiler policy, top of the page, answers in spoilers please.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

StevenJ
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### Re: Three princesses

Crosby wrote:(Please pardon me if this was already said.)

A simple solution:
Spoiler:
Ask "Which one of your sisters lies more (than sometimes)?"
Then pick the one she points to.

Let me explain:
If we eliminate as a candidate the sister we pose the question to, then, if we ask:
truth-teller - she points to the liar (success)
flipper - she points to either the liar or the truth-teller (success)
liar - she points the truth-teller (success)

That's not a yes/no question, but it's pretty easy to turn it into one that works:

Spoiler:
Call the three women Alice, Betty and Carol. Ask Alice, "Does Betty lie more often than Carol?" If yes, marry Betty. If no, marry Carol.

Crosby
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### Re: Three princesses

Sorry for lack of "spoiler" in original post.
(noob)

Also, FWIW - nothing in the original begs a yes/no question. Ask any question of one princess. Answer will be either truthful, or false... falseful?

thecaptainzap
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### Re: Three princesses

So I haven't bothered to read the 7 pages of posts. Sorry, I'm not THAT bored at work. I think I have a solution, assuming that you don't have a preference which princess you marry (as long as it is not the middle one).

Spoiler:
You ask each princess if she is, in fact, a princess.

The eldest will answer with a "Yes."
The youngest will answer with a "No."
The middle princess with answer with either a "Yes" or a "No."

So, you will either have 1 yes and 2 no, or 2 no and 1 yes. If you pick the princess who gave the single answer, then you either have the eldest or youngest. You'll even know which one you picked.

Let me know what you think.

Crosby
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### Re: Three princesses

well... I think the problem allows you to ask one question once (not 3 times, and not once per princess).

thecaptainzap
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### Re: Three princesses

And that would be the hole in my argument...

Oh well!

TDuck
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### Re: Three princesses

Really, it would seem that
Spoiler:
The relative age of the sisters is extraneous information

because
Spoiler:
there is a direct relationship between age and truthfulness,

which is to say that
Spoiler:
asking sister A if sister B is younger than sister C is exactly as successful as asking sister A if sister B lies more often than sister C.

In any case, you're hosed if you pick the liar:
"Do you take this man to be your lawfully wedded husband?"
"No."

And it'll just be annoying if you screw up and pick the mischief maker:
"Do you take this man to be your lawfully wedded husband?"
"No. Wait! Ask me again."

StevenJ
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### Re: Three princesses

Crosby wrote:Sorry for lack of "spoiler" in original post.
(noob)

Also, FWIW - nothing in the original begs a yes/no question. Ask any question of one princess. Answer will be either truthful, or false... falseful?

From the original post: "...the King will only grant you ONE yes or no question which you may only address to ONE of the sisters."

Crosby
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### Re: Three princesses

wow. I'm amassing noob-points is a hurry.
leet noob?

sorry for adding to the confusion on that, thanks for point it out.

z4lis
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### Re: Three princesses

Sorry if this has already been suggested. I don't feel like reading through seven pages. :\

Spoiler:
Can you not simply ask a question that cannot be answered? For instance, I whip out Schrödinger's cat and ask "Is the cat in this box dead?", the truthful and deceitful princesses cannot answer my question because they might provide a false or correct answer, respectively. The middle princess, however, doesn't care and will blurt out any answer.

EDIT: Here's a much better answer:

Spoiler:
Assuming all three princesses know a good bit of logic, just pick a statement that is always true, call it P, and any two girls, called 1 and 2, and ask 1 the question: "P if and only if 2 is consistent?" (By "consistent" I mean "always gives the same answer, true or false".) If the answer is "yes", choose 2. If the answer is "no", choose the third.

Why? A simple book in logic will explain that "A if and only if B" is equivalent to "(If A then B) and (If B then A)" which is equivalent to "(A or not B) and (B or not A)". First, note that if girl 1 is the inconsistent one, I don't care what her answer is. Either girl I choose is consistent. Let Q represent "2 is consistent". Now we have the other following cases:

Case 1: Girl 1 is the truth-teller, and Girl 2 is the liar. Girl 1 will answer P as true and Q as true. Therefore, she will respond to my question with "yes" and I pick Girl 2, who is the liar.

Case 2: Girl 1 is the truth-teller, and Girl 2 is the inconsistent one. Girl 1 will answer P as true and Q as false. Therefore, she will respond to my question with "no" and I pick the third girl, who is the liar.

Case 3: Girl 1 is the liar, and Girl 2 is the truth-teller. Girl 1 answers P as false and Q as false. Therefore, she will respond to my question with "yes" and I pick Girl 2, who is the truth-teller.

Case 4: Girl 1 is the liar, and Girl 2 it the inconsistent one. Girl 1 will answer P as false and Q as true. Therefore, she will respond to my question with "no" and I pick the third girl, who is the truth-teller.

In any case, I am guaranteed to not pick the inconsistent girl.

EDIT2: My solution actually has a gaping hole in it. Sigh.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

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### Re: Three princesses

Would this work?
Spoiler:
Q: (Are you not the middle sister) Xor (Are you telling the truth)?
Note: I used xor to make it easier to understand.
Sister 1: True xor True = false.
Sister 3: not(true xor false) = false.
Sister 2 when telling truth: False xor true = true
Sister 2 when lying: not(false xor false) = true

antonfire
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### Re: Three princesses

In the usual interpretation of the puzzle (the one I like best), sister 2 can say whatever the hell she wants. In other words, she could easily say "false" even to a question which would by some silly self reference seem to force her to say "true". Remember, the reason you don't want to marry her is that she's unpredictable.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

skeptical scientist
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### Re: Three princesses

Grenade: that sort of works, but there's a better solution which works without invoking any liar-paradox-ish statements.

As Anton said, think of the puzzle as saying, "the middle sister is mischievous and can give any answer to any question." Now liar-paradox-ish statements don't work, but there's still a solution.
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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GoldenYears
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### Re: Three princesses

Not that I've read all seven pages, as the solution was given on page 2, but maybe this'd work:
Spoiler:
'Do you have an elder sister that always tells the truth, and a younger that always lies?
If asked it's the eldest sister, she'd want to answer both no (no elder sister) and yes (younger sister that lies)
If asked it's the younger sister, she'd want to say no (to lie about her elder sister that tells the truth) and yes (because she doesn't have a younger sister that always lie)
Assuming the two above can't give a solid answer, the middle sister is the only one that will give either yes/no answer depending on whether she's lying/telling the truth at the time, because she can deny/accept both conditions.

VectorZero
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### Re: Three princesses

GoldenYears wrote:Not that I've read all seven pages, as the solution was given on page 2, but maybe this'd work:
Spoiler:
'Do you have an elder sister that always tells the truth, and a younger that always lies?
If asked it's the eldest sister, she'd want to answer both no (no elder sister) and yes (younger sister that lies)
If asked it's the younger sister, she'd want to say no (to lie about her elder sister that tells the truth) and yes (because she doesn't have a younger sister that always lie)
Assuming the two above can't give a solid answer, the middle sister is the only one that will give either yes/no answer depending on whether she's lying/telling the truth at the time, because she can deny/accept both conditions.

Except that they would parse this as
Spoiler:
Do you (have an elder sister that always tells the truth) AND (have a younger that always lies?)

The eldest would truthfully answer no.
The middle would say yes if truthful and no if lying.
The youngest would lie and answer yes.

Spoiler:
Eldest: (no) AND (yes) = no.
Middle: (yes) AND (yes) = yes, but may lie, or may lie in one part of the question, depending how you interpret the rules.
Youngest: NOT((yes) AND (no)) = yes
Van wrote:Fireballs don't lie.

Zalzidrax
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### Re: Three princesses

This is not a very pretty question, but I think it works:

Spoiler:
If you are not the eldest daughter, and if you are answer yes, is the middle sister answering this question with a lie?

The eldest replies yes, the youngest lies and says yes, but the middle must answer the question with a 'no' because answering yes would be paradoxical and neither a lie nor a truth.

jestingrabbit
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### Re: Three princesses

Zalzidrax wrote:This is not a very pretty question, but I think it works:

Spoiler:
If you are not the eldest daughter, and if you are answer yes, is the middle sister answering this question with a lie?

The eldest replies yes, the youngest lies and says yes, but the middle must answer the question with a 'no' because answering yes would be paradoxical and neither a lie nor a truth.

The most interesting interpretation of the question has the middle sister answering in a completely unstructured manner, so that they might answer any question with any answer and not break the rules. Your interpretation requires a more structured middle sister, and works for that case, but the other interpretation, and your own, have a really nice little answer.

Not very helpful hint:
Spoiler:
think about questions that refer to properties of the sisters.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

douglasm
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### Re: Three princesses

Assume the worst version of the middle sister: she completely ignores your question and (secretly) flips a coin to determine her answer.

Hint 1:
Spoiler:
How much information do you get from the answer?

Hint 2:
Spoiler:
It is possible to get no information from your question and answer whatsoever, yet you must still choose without asking again.

Hint 3:
Spoiler:
You don't care which of the non-middle sisters you get, you only need to eliminate the middle sister

Hint 4:
Spoiler:
How can you ensure that, if you gain no information at all, it doesn't matter?

Hint 5 (dead giveaway):
Spoiler:
Decide beforehand that you will not marry the sister who answers your question. Use the question only to decide between the other two.

brimilit
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### Re:

aaronspook wrote:Actually, my solution only relies on the sisters knowing their relative ages. Xandah's solution also works, as does any that effectively involves comparing an attribute that scales from the lying sister to the truth-telling sister. As far as I can think of, age and honesty are the only two such attributes we know of in this case.

If they have names that can be sorted according to an alphabetical scale, then asking A if B's name is earlier in the alphabet than C's will also work.

But if they do not know their ages, or their names, or their truthiness (or perhaps they just can't distinguish each other any more than the questioner can), there is still a way to ask that one question.

Spoiler:
Provided they are lined up before you, ask sister A (who is not in the middle) "is sister B physically closer to you than sister C"

Buttons
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### Re: Re:

brimilit wrote:
Spoiler:
Provided they are lined up before you, ask sister A (who is not in the middle) "is sister B physically closer to you than sister C"

Huh? That doesn't tell you anything more than "Is 1+1 equal to 2?" would, and clearly that question isn't enough. A "yes" could come from the oldest or the middle, and a "no" could come from the middle or youngest.

Rob7045713
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### Re: Three princesses

Spoiler:
"Would you say that the honest sister would say that you aren't the middle sister?"

Rob7045713
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### Re: Three princesses

Spoiler:
"If you were the lying sister, would you say that the honest sister would say that you are the middle sister?"

douglasm
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### Re: Three princesses

Neither of those questions gives you enough information. You would learn either that the sister you asked is not the honest sister or that the sister you asked is not the lying sister. What you need to know is one sister that is definitely not the middle sister.

Read my hints a few posts back if you're having trouble.

zombieshoju
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### Re: Three princesses

Hi this is my first post so please be kind I think I have a solution it dose however assume that all three sisters are in the same room and don't move win you are asking questions.

Spoiler:
you ask the middle sister is the sister to your left able both lie and tell the truth exclusive or are you a liar .
You then marry the one on the left if no and the one on the right if yes

Spoiler:
all possible situations
left center right part 1 part 2 truth persons answer who to marry
truth liar 50/50 n y = y = n marry left truth
50/50 liar truth y y = n = y marry right truth
liar truth 50/50 n n = n =n marry left liar
50/50 truth liar y n = y = y marry right liar
liar 50/50 truth marry either left or right
truth 50/50 liar marry either left or right
never marry the center sister.

douglasm
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### Re: Three princesses

I believe that is a correct answer. Just out of curiosity, how many, if any, of my hints did you read before coming up with that?

afarnen
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### Re: Three princesses

"Are you still beating your wife?"

fabiocbinbutter
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### Re: Three princesses

*hand her a pencil and paper* "Is 31337 a prime number?"... After a few hours, if you don't have your answer, you're safe.

suicyclist
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### Re: Three princesses

I love this problem.
Strangely though, i like the nasty complicated answers better than simple ones.
maybe i should be a programmer.

Spoiler:
"If you are not the middle sister would the other non-middle sister answer truthfully, or if you are the middle sister and this total question were instead 'are you lying', would your answer be valid?"

as in "say yes if[condition 1]OR[condition 2]"

the first part is designed to force the answer to 'no' if the question is directed at the oldest or youngest; the second part draws a yes answer from the middle sister regardless of her mindset. 'are you lying' is invalid if she really is lying, so the truthful answer would be no, so the actual answer will be yes. this lets me get away with using a question that by itself is not allowable.

AvalonXQ
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### Re: Three princesses

suicyclist wrote:I love this problem.
Strangely though, i like the nasty complicated answers better than simple ones.
maybe i should be a programmer.

Spoiler:
"If you are not the middle sister would the other non-middle sister answer truthfully, or if you are the middle sister and this total question were instead 'are you lying', would your answer be valid?"

as in "say yes if[condition 1]OR[condition 2]"

the first part is designed to force the answer to 'no' if the question is directed at the oldest or youngest; the second part draws a yes answer from the middle sister regardless of her mindset. 'are you lying' is invalid if she really is lying, so the truthful answer would be no, so the actual answer will be yes. this lets me get away with using a question that by itself is not allowable.

Again, you're assuming that the middle sister goes into the question saying "I'm going to lie this time" or "I'm going to tell the truth this time". The "preferred" interpretation is that the middle sister can answer however she wants. In other words, she doesn't even have to listen to your question if she doesn't want to; she can say "no" anyway.

Goldstein
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### Re: Three princesses

Yes, it's probably best to assume that they're princesses, not logicians. As such, they won't be running your gauntlets of logic.

I don't appreciate being told that I want to marry out of 'knowing where I stand' though; clearly I pick the best looking of the three.
Chuff wrote:I write most of my letters from the bottom

jestingrabbit
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### Re: Three princesses

The Duchess Lovelace was such a fine logician that one of the first computer programming languages, ada, is named after her. I can't imagine why a princess couldn't be a logician.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Goldstein
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### Re: Three princesses

Alright, there was no need for me to emphasize 'princesses' but I do think this puzzle works best without assuming that the middle sister is tied to any sort of routine. Otherwise, it's just a question of setting up something akin to that which suicyclist suggested, which, whilst clever, isn't nearly as elegant.
Chuff wrote:I write most of my letters from the bottom

jestingrabbit
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### Re: Three princesses

I agree, the more interesting interpretation involves a truly random middle sister, one who can answer in any way to any question.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

prokrastinatorSF
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### Re: Three princesses

Don't know if this has been posted yet but it's different than the accepted best answer, which I have to admit is genius. I was coming at this from the background of the "crossroads" puzzle so I approached it in a similar way, and as such ignored the "no meta-questions" rule. See what you think.

Spoiler:
Ask any sister: "Would the sister who always answers the way you are answering now say yes if asked if you are the middle sister in age?"
*The truth-telling sister would say no. She is the sister who always answers the same way she is answering, and she would reply that she is not the middle sister.
*The lying sister would say no. She is the sister who always answers the same way that she is answering, she is not the middle sister, therefore if asked if she is the middle sister she would reply falsely that she was the middle sister, but she would lie about this hypothetical and answer.
*The middle sister would say yes. If she were telling the truth, she would say that the older one WOULD identify her as the middle sister, and she would report this. If the younger sister were asked, she would falsely say that the middle sister was NOT the middle sister, but since in reporting the younger sister's answer she lies, the middle sister would answer "yes."

It was clearer in my head than in this post. I think, at least, that it is better than asking "what would YOU answer" to this question because then the middle sister could conceivably answer the meta-question falsely but the internal question truthfully, or vice-versa.

Also, the middle sister should stab people who ask tricky questions.