## Three princesses

A forum for good logic/math puzzles.

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jestingrabbit
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### Re: Three princesses

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

There's a way to make the problem a little trickier by making the middle sister completely unguided by the question. In this way tricky questions become harder to use. There is a very simple question which works.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

simdude
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### Re: Three princesses

Spoiler:
"If you asked your sisters if you are the youngest would they HAVE to agree?"

poss.1: you asked the oldest. The youngest would would have to say yes. The middle could say yes or no. The oldest would be telling the truth in that they do not HAVE to agree.
poss.2. you have asked the middle. The youngest would have to say yes. The oldest would have to say no. They do not agree. The middle
poss.3 you have asked the youngest. The oldest would say yes. The middle would say yes or no. They do not have to agree. The youngest has to lie and say they WOULD agree.

I think this works?

Edited to throw out missing something in my own thought process...

I didn't want to read too many of the previous pages to get too many ideas. But is the secret to this an objective question followed by how would you sisters respond. Or how would your sisters say you would respond? I'm just missing it...

douglasm
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### Re: Three princesses

There is a single core principle you have to figure out, and once you know it you should easily be able to construct a suitable question. I posted a set of hints for it on the previous page, quoted here for convenience. The first will do no more than try to start you thinking down the right path, and the rest go progressively further.
douglasm wrote: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.

Rangsk
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### Re: Three princesses

This has already been elegantly solved, however I came up with a decidedly unelegant and cheap solution which I'd like to share.

Spoiler:
This, of course, assumes you have time to prepare and address the three sisters with a nice speech (a "get to know me" as it were) before asking your question. It then exploits the fact that only the middle sister has a choice of what to answer. The other two sisters must answer "yes" or "no" based on the constraints of truth, rather than whim.

So, for the preparation: Dress yourself in something very ugly and low class. Don't shave or bathe for a while. Basically, make yourself as unsightly as possible.

Now, the speech: Explain to the women that after marrying one of them, you will be horrible to her. Be creative, but basically the idea is that these women will not want to marry you, under any circumstances. Then, explain that the following question is designed such that someone who always tells the truth or someone who always lies will come up with the same answer... that answer being "yes." Someone who does not can choose, if she wishes, to answer "no." Also, explain that your intention is that you do not wish to marry the sister who can choose.

The question: Are you the sister who always tells the truth?

Then, of course, if the sister you asked answers "yes" then marry her. If the sister you ask answers "no" then marry someone else.

The only problem I see with this solution is that the preparation or speech may make the king unhappy and he would call off the whole thing. If that is a worry, then perhaps you could be a little more subtle about making the sisters not want to marry you. I'm sure it wouldn't be that hard. It turns into a problem of psychology, rather than logic.

Token
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### Re: Three princesses

What if the middle sister likes that kind of thing?
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

Zaide
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### Re: Three princesses

You cannot win with this problem however you can make your increase by acting a very simple question. "Which one of these lovely ladies is your sister.
Truth Sis: They are both my sister
Lie Sis: Neither are my sister
Middle Sis: She is my sister

jestingrabbit
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### Re: Three princesses

Zaide wrote:You cannot win with this problem however you can make your increase by acting a very simple question. "Which one of these lovely ladies is your sister.
Truth Sis: They are both my sister
Lie Sis: Neither are my sister
Middle Sis: She is my sister

Firstly, you may ask a yes or no question only. Secondly, I don't think the way that you describe the middle sister answering makes any sense. She could say anything. Thirdly, there is a very simple question that allows you to determine that one of the sisters is not the middle sister, and that is all you need to do.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

josiahstevenson
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Contact:

### Re: Three princesses

I bet this has been suggested already, but I want to get it in before reading all of them

Spoiler:
You ask, "does your older sister tell the truth exclusively?"

Eldest: I have no older sister
Middle: Yes or No
Youngest: "which one?"

if you get a straight answer from the girl you ask, you marry either other one.

jestingrabbit
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### Re: Three princesses

Asking a question that isn't answerable with a yes or no for every sister isn't allowed.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

kansasdave
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### Re: Three princesses

Spoiler:
You ask: "Is the middle princess going to answer this truthfully, or are you the youngest princess?

If asked to the eldest, the true answer is 'no,' so she says 'no.'
If asked to the youngest, the true answer is 'yes,' so she says 'no.'

If asked to the middle princess, if she says 'yes' then the true answer is 'yes,' but there is no false answer because 'no' isn't false, just a paradox.

So you ask the question of one of the girls, marry her if she says 'no,' and marry one of the others if the one you asked says 'yes.'

kansasdave
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### Re: Three princesses

Spoiler:
Right. Ask A whether B would say C is the youngest. If you're asking the eldest and C is the middle, she'll say 'yes'; if you're asking the youngest and C is the middle, she'll say 'yes.'

So ask the question, if you get a 'yes' marry B, and if you get a 'no' marry C. If A is the eldest or youngest then this avoids marrying the middle, and if A is the middle it doesn't matter how she answers since you're not marrying A in either case.

jestingrabbit
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### Re: Three princesses

kansasdave wrote:
Spoiler:
You ask: "Is the middle princess going to answer this truthfully, or are you the youngest princess?

If asked to the eldest, the true answer is 'no,' so she says 'no.'
If asked to the youngest, the true answer is 'yes,' so she says 'no.'

If asked to the middle princess, if she says 'yes' then the true answer is 'yes,' but there is no false answer because 'no' isn't false, just a paradox.

So you ask the question of one of the girls, marry her if she says 'no,' and marry one of the others if the one you asked says 'yes.'

This requires the middle princess to answer in a structured way. The more interesting interpretation of the puzzle arises when the middle sister is answering in a completely random way, completely disregarding the question.

kansasdave wrote:
Spoiler:
Right. Ask A whether B would say C is the youngest. If you're asking the eldest and C is the middle, she'll say 'yes'; if you're asking the youngest and C is the middle, she'll say 'yes.'

So ask the question, if you get a 'yes' marry B, and if you get a 'no' marry C. If A is the eldest or youngest then this avoids marrying the middle, and if A is the middle it doesn't matter how she answers since you're not marrying A in either case.

This is a lot close to the best answer, but there is improvement possible.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

kansasdave
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### Re: Three princesses

Spoiler:
Ask A: "Is it true that A is the eldest and B is the middle, or that A is the youngest and B is the eldest?"

Case: A eldest, B middle: answer is true, A says 'yes'
Case: A youngest, B eldest: answer is true, A says 'no'
Case: A eldest, B youngest: answer is false, A says 'no'
Case: A youngest, B middle:answer is false, A says 'yes'

So if the answer is 'yes' then C is definitely not the middle princess, and if the answer is 'no' then B is definitely not the middle princess.

jestingrabbit
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### Re: Three princesses

Go for something simpler. There is a very simple, entirely uncomplicated answer.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Philosiraptor
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### Re: Three princesses

Ask the King, it never said the question was to be directed to the princesses.

douglasm
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### Re: Three princesses

Yes it did, and the simple answer does not involve any such loophole in the problem statement.

kansasdave
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### Re: Three princesses

Spoiler:
Ask A: 'If asked, would you confirm that B is the middle princess?'

If B is in fact the middle princess, the eldest and youngest would both say 'yes.' If B is not the eldest, they would both say 'no.' Marry C if you get a 'yes' and B if you get a 'no.'

EnderSword
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### Re: Three princesses

kansasdave wrote:
Spoiler:
Ask A: 'If asked, would you confirm that B is the middle princess?'

If B is in fact the middle princess, the eldest and youngest would both say 'yes.' If B is not the eldest, they would both say 'no.' Marry C if you get a 'yes' and B if you get a 'no.'

Spoiler:
Why would the Youngest answer 'Yes' if B was the Middle Princess? She'd Lie and say 'No'
And if B was not the eldest, then why would both youngest and oldest say 'No'?
This isn't correct.
WWSD?*
*what would Sheldon do?

AvalonXQ
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### Re: Three princesses

EnderSword wrote:
kansasdave wrote:
Spoiler:
Ask A: 'If asked, would you confirm that B is the middle princess?'

If B is in fact the middle princess, the eldest and youngest would both say 'yes.' If B is not the eldest, they would both say 'no.' Marry C if you get a 'yes' and B if you get a 'no.'

Spoiler:
Why would the Youngest answer 'Yes' if B was the Middle Princess? She'd Lie and say 'No'
And if B was not the eldest, then why would both youngest and oldest say 'No'?
This isn't correct.

Spoiler:
Actually, it is correct.
If B is the middle princess, then the Youngest would NOT confirm that B is the middle Princess.
So, if you ask her if she would confirm it, she'd lie and say "Yes".
Similarly, if B is not the middle princess, then the Youngest WOULD confirm that B is the middle Princess if asked. So she'll lie to the question asked and say "No".
Essentially, the "if asked, would you" question creates truth by double negation for liars.

EnderSword
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### Re: Three princesses

Spoiler:
Gotchya, I misread that as 'If you asked A...' I see that's actually meant to be part of the question.
WWSD?*
*what would Sheldon do?

Greyson
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### Re: Three princesses

But you might ask the middle child.

OK - away from the subject of the puzzle itself: Remind me. Why do I want to go to so much effort to marry a girl who has at most a 2 word vocabulary with a 2/3 chance of choosing a liar or a prankster. Even the third girl must be kinda messed up in the head having to ALWAYS answer truthfuly AND having had to live with the other girls for at least the last 18yrs (I would hope)

Anyway - back on topic.

Here's how I see it:
Talk to the eldest or youngest and you can force a yes or a no answer depending on how you phrase your question
Talk to the middle and they have a 50:50 chance of giving you that same answer and an equal chance of giving an answer that proves that she is neither the oldest or youngest

This gives you a 1:6 chance of weeding out the middle girl - all other answers are indefinitive

xkcd - is there actually a definitive answer to this? Are you sure your solution can't be pulled apart because of the unpredictability of the middle childs answer?

kansasdave
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### Re: Three princesses

Greyson wrote:But you might ask the middle child.

Spoiler:
Asking the middle child isn't a problem. If you ask the middle child (so she's A) then neither of the princesses you might marry (B or C) is the middle princess.

douglasm
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### Re: Three princesses

Greyson wrote:Here's how I see it:
Talk to the eldest or youngest and you can force a yes or a no answer depending on how you phrase your question

Spoiler:
You could force a 'yes' or force a 'no', but that's not very useful. You can instead force an answer that is independent of which of the two you asked and determines one sister who is not the middle sister.

Greyson wrote:Talk to the middle and they have a 50:50 chance of giving you that same answer and an equal chance of giving an answer that proves that she is neither the oldest or youngest

This gives you a 1:6 chance of weeding out the middle girl - all other answers are indefinitive

Spoiler:
You're taking the wrong approach. First, deal with the possibility of talking with the middle girl. Doing so is very simple, just decide that you will marry one of the two you don't talk to. The middle girl's answer no longer matters. Second, design a question that will get the same response from both the oldest and youngest girls and will tell you which of the other two to avoid.

Greyson wrote:xkcd - is there actually a definitive answer to this? Are you sure your solution can't be pulled apart because of the unpredictability of the middle childs answer?

Yes. Several such solutions have been posted and exhaustively checked.

JustJohn
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### Re: Three princesses

Before reading the thread this is the best I can come up with:

Spoiler:
"Do either of your sisters always lie?"

This has the advantage of getting the same answer (yes) from both the sisters you want. Assuming the middle sister lies half of the time you'd get an 83% success rate (5/6), but you could of course have the bad misfortune to ask the middle sister and have her tell the truth and then you wouldn't know it was her.

EDIT1: Read most of the thread now. Fun problem.

Spoiler:
The whole asking if one sister is "more truthful" is super clever because the result gives you the second piece of information that we would have resorted to a meta to get otherwise. Nice.

lordlicorice
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### Re: Three princesses

My analysis:

Like Shoofle, I initially balked at the idea that this problem is possible. You need to ask at least two questions to decide the problem (obviously the second question needs to be conditional or else they could be linearly combined to one long question). You need two bits of information (yes/no answers) and the problem specifically restricts you to one: the problem is impossible.

But it is possible. I want to reassure the noble logic warriors valiantly trying to solve this on their own that there is a simple, logical answer with no weird trickery involved.

Spoiler:
Of course, I didn't notice that you actually only need one bit of information because the younger/middle/older relational structure lets you identify someone with "less-than and greater-than" rather than "equals." This intuitively makes sense because this "extra information" lets you bypass the Shannon entropy of the second bit, or provides the substantive mapping from one bit to two, depending on how you look at it.

Try graphing which sister you should decide on, based on the answer to the question "Which of your sisters can lie?" On one axis put the answer to the question, on another put whether or not the respondent is lying. Notice that you cannot linearly separate your decision, and in fact you have a third orthogonal axis with a variable of uncertainty since you don't know who to pick if the respondent is telling the truth.

Now graph the question "Which of your sisters is younger?" with "Telling truth? Y/N" on one axis and "Answered? Sister A / Sister B" on another, with an A/B for each combination signifying whether you should pick A or B. Notice that the decision procedure is linearly separable: simply check on which side of the line between the halves of Answered? you are. Or in other words, you just pick the same value as "Answered?"

Note that linear separability is not enough because a decision-line is two-variable: a slope and y-intercept. If you graphed it like I did, then your one bit of input only gives you a y-intercept so the decision line must be horizontal. The trick is to come up with a question that results in a horizontal decision line. The relationship between the sisters is the obvious tool to use. The correct question falls into place easily when the problem has been explored to this point. 20/20 hindsight.

I was just reading Minsky's perceptron paper and after playing around with those graphs was struck by linear separability as a way to explain these sort of logic puzzles. Hope it's not too trivially obvious for the geniuses who came up with the solutions in this thread.

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### Re: Three princesses

Spoiler:
would both of your sisters give me the same answer to this question?
True: "don't know." b/c she has no idea what the random sister will answer
Random: "yes" or "no" depending on whether she lied or told the truth
False: "don't know"

However this doesn't work if the question is:
Of the three sisters, one always lies, one always tells the truth, and one can answer in any way she wants. Thus if the middle sister decides that she likes you she could say "don't know" rather than being restricted to a yes/no answer.

lordlicorice
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### Re: Three princesses

Spoiler:
would both of your sisters give me the same answer to this question?
True: "don't know." b/c she has no idea what the random sister will answer
Random: "yes" or "no" depending on whether she lied or told the truth
False: "don't know"

However this doesn't work if the question is:
Of the three sisters, one always lies, one always tells the truth, and one can answer in any way she wants. Thus if the middle sister decides that she likes you she could say "don't know" rather than being restricted to a yes/no answer.

Of course if she's lying she would say she does know.

paulrowe
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### Re: Three princesses

Not caring for the difficulty of actually trying to phrase the question posed below, I opted to find a different one.
zombieshoju wrote:
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:
Ask the following question of the sister in the middle: "Would you agree with the sister to your left that you are the eldest?"

If I ask this question of the eldest, her inability to predict the answer of the middle sister would immediately clue me in to which one she is. If she answers, "I don't know," the sister to her right is the youngest; otherwise, she says, "No," and the sister to her left is the youngest.

If I ask this question of the youngest, again, her inability to predict the answer of the middle sister should also clue me in to which one she is. If she can't give me a "Yes" or "No" (how would she answer?), the sister to her right is the eldest; otherwise, she says, "No," and the sister to her left is the eldest.

In any case, I won't choose the sister to whom the question is addressed, so asking the question of the middle sister will result in a favorable outcome regardless of her answer. Getting a straight "Yes" or "No" answer would lead me to pick the sister to her left, and anything else would lead me to pick the sister to her right.

This is dangerously close to a meta-question (and was originally a meta-question), but I find it a lot easier to phrase. Comments?

dedalus
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### Re: Three princesses

Thanks for the spoilers JR; I'm 99% sure you've jumped onto the same solution I did.
Here's a hint for people still stuck (probably a rewording of previous hints):
Spoiler:
You're trying to find A princess which isn't the middle princess. Not whether the princess you question is the middle princess.
Oh, and the correct question (bar rewording) is 6 words long.

Judging by the way the mods are acting here I'm guessing it's out of place for me to post the answer.
doogly wrote:Oh yea, obviously they wouldn't know Griffiths from Sakurai if I were throwing them at them.

douglasm
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### Re: Three princesses

dedalus wrote:
Spoiler:
Oh, and the correct question (bar rewording) is 6 words long.

Spoiler:
It can be done quite well in 5 words, actually.

dedalus wrote:Judging by the way the mods are acting here I'm guessing it's out of place for me to post the answer.

Post as many variations on the answer as you like, as long as they're all in clearly labeled spoilers. For example, here's the simplest solution I recall:
Spoiler:
With the sisters labeled A, B, and C, ask A "Is B older than C?" If A says yes, marry C. If no, marry B.

Posting explicit answers is only bad if you do it in a way that makes it easy for people who don't want to be spoiled to read the answer by accident.

6453893
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### Re: Three princesses

Read the first one first.
Spoiler:
Not a solution, but an extension of the question. Warning: Assumes you've read the answer already.

Spoiler:
It bothers me that the solution depends on you not marrying the sister to whom you address the question. Saying "If you ask the middle sister, it doesn't matter because you can marry either of the other two" seems like cop out to me. What if you wanted to marry the middle sister? Is there a solution guaranteed to find the middle sister?

jestingrabbit
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### Re: Three princesses

@6453893 re your second question.
Spoiler:
No. Consider that you have only one bit of information after the question is answered and that you would need either a trit or two bits or more information to decide between three possibilities. Its a good question though, it helped me to arrive at the answer.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Posts: 192
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### Re: Three princesses

easy:
Spoiler:
ask is the sky blue?
choose the princess whose answer is different from the other two.
I... I didn't do it.
<- he did it, I swear

douglasm
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### Re: Three princesses

Spoiler:
ask is the sky blue?
choose the princess whose answer is different from the other two.

You only get an answer from one of them, not all three.

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### Re: Three princesses

dang. oh well, worth a shot
I... I didn't do it.
<- he did it, I swear

rddrgn
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### Re: Three princesses

(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

jestingrabbit
Factoids are just Datas that haven't grown up yet
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### Re: Three princesses

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.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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### Re: Three princesses

Spoiler:
Without looking at any of the answers, I'd say this
'Which of these sisters are older than you?'
oldest says none
youngest says none
middle lying says youngest
middle truthing says oldest

jestingrabbit
Factoids are just Datas that haven't grown up yet
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### Re: Three princesses

Spoiler:
Without looking at any of the answers, I'd say this
'Which of these sisters are older than you?'
oldest says none
youngest says none
middle lying says youngest
middle truthing says oldest

That's a good try, but the question can only be answered "yes" or "no". You are definitely on the right track though.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Hydralisk
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### Re: Three princesses

Spoiler:
"Would both your sisters give a consistant answer if I asked them this question repeatedly?".

If you've asked the eldest, she will say no, and you can marry her.
If you've asked the youngest, she will say no, and you can marry her.
If you've asked the middle one, and she says yes, marry one of the other ones.
If you've asked the middle one, and she says no, you marry her, get divorced/have her killed and re-marry.

This solution gives a 3/4 chance of marrying the Eldest/youngest sister.