## Three princesses

A forum for good logic/math puzzles.

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ggrot
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I just realized that this is pretty similar to karmadillo's answer. The key seems to be making sure that irregardless of the answer to your question, you don't select the sister to whom you asked the question. This eliminates any concern about getting a "random" answer because if you did get a random answer, you win via your strategy.

ggrot
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Karmadillo's answer is slightly wrong because the middle-aged sister, while able to lie, may never choose to do so in which case she doesn't lie more than the truth-telling sister. Change the question to something like "Ask sister 1 if sister 2 is older than sister 3" and it should work flawlessly.

EmperorcJ
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Here's a solution that doesn't involve relative ages.

Ask sister #1 if this statement is true: #2 is capable of lying if-and-only-if #3 always lies

I made a little table here: (* for middle sister, T for one who always tells the truth, F for one who always lies)

Code: Select all

`1 2 3-----T F * no     (T iff F -> F -> no)  T * F yes    (T iff T -> T -> yes)* F T yes/no * T F yes/noF * T yes    (T iff F -> F -> yes)F T * no     (F iff F -> T -> no)`

If 'no' marry sister #2, if 'yes' marry sister #3

monkpalmer
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'Did the middle sister think about polar bears over breakfast this morning?'

Only the middle sister will know the answer. She is mischievous, so will answer either "Yes" or "No" according to her whim.

Because they won't know the answer, neither the eldest nor the youngest will reply. In order to lie or to tell the truth you need to know what the truth is.

So if the sister you ask is silent, go ahead and marry her. If she answers "yes" or "no", marry one of the others.

bhauth
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[ Are you lying? ] XOR [ Does one of the others switch her responses? ]

true -> no XOR yes -> yes
false -> yes XOR yes -> no -> yes
either: true -> no XOR no -> no
either: false -> yes XOR no -> yes -> no

Minkoff
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So, before reading anything but the first post, I came up with this:

Do you have an older sister who would ever answer a yes or no question with the same level of truthfulness that you are now?

Ask for a different one if she says yes.

The reason I came up with such a weird answer is that I assumed you couldn't have all the sisters in the same room together. I'm tired though. Can someone smarter/more alert than me confirm or disprove this one?

skeptical scientist
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monkpalmer wrote:What if you ask this:

'Did the middle sister think about polar bears over breakfast this morning?'

Only the middle sister will know the answer. She is mischievous, so will answer either "Yes" or "No" according to her whim.

Because they won't know the answer, neither the eldest nor the youngest will reply. In order to lie or to tell the truth you need to know what the truth is.

So if the sister you ask is silent, go ahead and marry her. If she answers "yes" or "no", marry one of the others.

What if, unbeknownst to you, the three of them discussed polar bears over breakfast this morning? Then all three of them would know the answer!
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

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bhauth wrote:[ Are you lying? ] XOR [ Does one of the others switch her responses? ]

true -> no XOR yes -> yes
false -> yes XOR yes -> no -> yes
either: true -> no XOR no -> no
either: false -> yes XOR no -> yes -> no

If you ask either the oldest or the youngest sister this question, then it is of the form "Are you lying XOR true", which is equivalent to the question "are you telling the truth". If she says "no", then the result is the liar paradox, and if she says "yes", then the result can be consistently interpreted as either true or false. If you ask the middle sister, you have the same possibilities (liar paradox or consistently interpretable as true or as false), but they are reversed.

The puzzle is not entirely clear what the sisters will do under these circumstances, so I'm not sure if this counts as a solution. However, you've designed it in such a way that there is one paradoxical answer and one always consistent answer for each, and for the middle sister the consistent answer is different from the other two, which is quite clever.
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

Maseiken
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monkpalmer wrote:What if you ask this:

'Did the middle sister think about polar bears over breakfast this morning?'

Only the middle sister will know the answer. She is mischievous, so will answer either "Yes" or "No" according to her whim.

Because they won't know the answer, neither the eldest nor the youngest will reply. In order to lie or to tell the truth you need to know what the truth is.

So if the sister you ask is silent, go ahead and marry her. If she answers "yes" or "no", marry one of the others.

But the youngest Knows that she doesn't know, so if she says "Yes" or "No" she is lying.
Q.E.D (I just like saying that)
"GRRRRRRRRRROOOOOOOOWR!!!!"
(Translation: "Objection!")

Maseiken had the ball at the top of the key...

SF
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Oldest sister can't tell a lie.
Youngest sister can't tell the truth.

Oldest sister must say "i don't know" or "probably" as there is a 75% chance they'll give the same answer.
Youngest sister can't answer yes/no or she might otherwise happen to tell the truth by mistake. She'll have to say "improbable".
Middle sister will still take this as a yes/no question, so whatever she answers you'll know it's her.

EmperorcJ
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I believe you have to ask a question where yes and no are the only possible answers, for it to be a yes or no question.

SF
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There is no such question. Anyone can evade the question if they want to.

Q: Is the sky blue?
A: Ask someone else, i'm busy.

charminggeek
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I believe I may have a solution. It works in my head, but I'm still in an early morning stupor.

Say we line the girls up, labeling them Left (L), Middle (M) and Right(R), relative to the girl in the center. Now we ask M, "Does your sister to the left (L) lie more often than your sister on the Right (R)?"
If she says "yes," marry L. If "no", marry R.

Care to verify?

charminggeek
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...or I could just read page three.

mmauz
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### maybe

I skimmed the 1st page & tried this:

Ansain
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sorry but I do not have the time to go through and see if this has allready been said, but I have to leave soon and want my answer to be heard I do believe this works

ask any one sister whether one of the other two sisters is more likely to lie to you. if you get a yes go with the sister you were asking about if you get a no go with the other one. never go with the sister you questioned. if you asked the liar, you will end up with the truth teller. if you asked the truth teller you will end up with the liar. if you asked the middle sister you have a 50% chance of either one both of which are good.

SF
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Ansain wrote:sorry but I do not have the time to go through and see if this has allready been said, but I have to leave soon and want my answer to be heard I do believe this works

ask any one sister whether one of the other two sisters is more likely to lie to you. if you get a yes go with the sister you were asking about if you get a no go with the other one. never go with the sister you questioned. if you asked the liar, you will end up with the truth teller. if you asked the truth teller you will end up with the liar. if you asked the middle sister you have a 50% chance of either one both of which are good.
You weren't asking about a specific sister; you were asking if one of them was more likely to lie to you.

dkfn
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### alternative solution?

Hi all,

I've got what I think is a valid solution but that doesn't quite match the spirit of the puzzle. Essentially, I added in a time element. So, the question would be: "When I point to your middle sister answer this question: 'Is the Sky Blue'?". -- sorry if anyone saw this when it wasn't white

Thoughts...?

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### Re: alternative solution?

dkfn wrote:Hi all,

I've got what I think is a valid solution but that doesn't quite match the spirit of the puzzle. Essentially, I added in a time element. So, the question would be: "When I point to your middle sister answer this question: 'Is the Sky Blue'?". -- sorry if anyone saw this when it wasn't white

Thoughts...?

I don't think you're allowed to instruct the sisters when to answer the question you ask.
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

Peripatetic
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I haven't read the rest of the discussion because I wanted to get my answer up without hints and just in case there is no correct answer yet.

Forgive my superficiality:

Walk up to the ugliest of the princesses. Point to another princess. Ask the first, "Is the princess I'm pointing at older than the third princess?"

If the answer is yes, marry the third princess. If the answer is no, marry the princess you're pointing at.

Five situations:

1) If you ask the question of the truth-teller and point at the liar, the answer is no. You marry the liar.

2) If you ask the truth-teller and point at the mischievous one, the answer is yes. You marry the liar.

3) If you ask the liar and point at the truth-teller, the answer is no. You marry the truth-teller.

4) If you ask the liar and point at the mischievous one, the answer is yes. you marry the truth-teller.

5) If you ask the mischievous one, it doesn't matter what the answer is since you never marry the princess you question.

Nuts, aaronspook used this at the top of page 2.

1000 posts and still no title
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Okay, the puzzle must've been solved by now, but I don't wanna cheapen my victory by reading through to see if I'm -really- right.

EDIT: I figured the problem had already been beaten to death, and I was probably wrong anyway, so a spoiler tag wouldn't be necessary. I was wrong on the internet .

spoiler wrote:Assuming that "I don't know" or no response are valid responses, and that the lying sister won't make up an answer if she doesn't know the answer:

You go to sister 1 and ask, "What would sister 2 say if I asked her what sister 3 would say if I asked her if she was the eldest?". If they can't answer, marry 'em.

If you're talking to the liar or the truthy one, they won't be able to accurately determine what flip-flopper will say, and so won't have a definite answer.
If you're talking to flip-flopper, she knows what the other two will say, and so can give a definite answer one way or the other.

EDIT: I also didn't realize that we're not allowed to explode their heads...
Last edited by Owijad on Mon Jun 18, 2007 6:16 pm UTC, edited 3 times in total.
And if you win you get this shiny fiddle made of gold,
But if you lose, the devil gets your sould!

jestingrabbit
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Firstly- spoiler policy!! Secondly, there are answers that don't rely on there being a "no possible answer" response. If you require such a response... it seems as though you're breaking the problem rather than solving it.

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Yeah, the OP didn't specify that he didn't depend on non yes/no answers, and I didn't want to read through his hints :/
And if you win you get this shiny fiddle made of gold,
But if you lose, the devil gets your sould!

Ansain
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Ansain wrote:
sorry but I do not have the time to go through and see if this has allready been said, but I have to leave soon and want my answer to be heard I do believe this works

Quote:
ask any one sister whether one of the other two sisters is more likely to lie to you. if you get a yes go with the sister you were asking about if you get a no go with the other one. never go with the sister you questioned. if you asked the liar, you will end up with the truth teller. if you asked the truth teller you will end up with the liar. if you asked the middle sister you have a 50% chance of either one both of which are good.

You weren't asking about a specific sister; you were asking if one of them was more likely to lie to you.

in my answer to make it a yes or no question you you have to be asking about a specific sister
"is she *points to one sister* more likely to lie to me than she is *points to other sister" the one you are asking about is the first one sorry not to specify that, I thought it was just assumed.
also, sorry to charminggeek who posted the same answer before I got to it. I hadnt read the entire topic yet.

theepicsnail
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So i dont know if this is too simple or if there are any weird cases , but why not simply ask:

if it's the oldest or the younges you'll get a "well ... molly and uh... n/a?"

and the middle sister lies (or tells the truth) but it doesn't matter because what's your sisters name , whether you lie or not about her name , it's not "not applicable" so she can lie or tell the trust about the names all she wants

jordan
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What a devious problem! This took me nearly quarter of an hour of doodling on some paper, but I got there in the end. Next stop, the three Gods problem!

Solution:

Go up to any daughter and point at one of her sisters. Ask, "Is she younger than your other sister?"

The eldest daughter will tell the truth, and say:
• "Yes" if you are pointing to the youngest daughter;
• "No" if you are pointing to the middle daughter.

The youngest daughter will lie, and say:
• "Yes if you are pointing at the eldest daughter;
• "No if you are pointing at the middle daughter.

The middle daughter will answer at random.

If the girl you asked says "Yes," pick the daughter you're pointing at. Otherwise, pick the other sister (i.e. the one you're not talking to or pointing at). You are guaranteed to never pick the middle daughter.

Some hints which might help others get on the right track without spoiling the problem completely (this is how I got my answer):

2. It soon becomes obvious that the problem is the middle daughter, who basically answers at random. Don't waste time trying to find a question which forces her to answer the same way whether she's lying or not; the solution comes when you figure out a way of ignoring her input. I got my answer by drawing a truth table with her column removed, and writing down the truth function I wanted my question to have to get a reliable answer.

If you're really, really stuck:

You want to design a question such that the oldest daughter will give you the same answer as the youngest daughter when you point at the middle daughter. You might want to ask about their relative ages...
Last edited by jordan on Wed Jul 11, 2007 1:53 pm UTC, edited 1 time in total.

skeptical scientist
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Jordan, your solution and hints are clearly visible (at least to me). General practice is to put them in quote boxes so that the background is the same color as the text (well, technically the background of a quote box is #FAFAFA, but this is close enough that you won't be able to read it). You might want to go back and fix this. It's not such a big deal here as we're on page 4 and solutions have already been discussed without hiding spoilers, but it's good for future reference.
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

jordan
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skeptical scientist wrote:Jordan, your solution and hints are clearly visible (at least to me). General practice is to put them in quote boxes so that the background is the same color as the text (well, technically the background of a quote box is #FAFAFA, but this is close enough that you won't be able to read it). You might want to go back and fix this. It's not such a big deal here as we're on page 4 and solutions have already been discussed without hiding spoilers, but it's good for future reference.

Thanks for that, skeptic; I've fixed it as you suggested and will bear it in mind for future reference. I did notice that I could read the text fairly easily, but I wasn't sure about the best way to go about hiding it. Forums deal with this sort of thing as a matter of course often have a special "spoiler" code which either disguises the text by matching the background colour, or sets the CSS display property to "none" and switches it back to "block" using Javascript. (The latter is particularly good because it prevents many screen-readers from accidentally reading spoilers aloud.)

Is there any chance that such a thing might be implemented for xkcd some time? It would be pretty useful in these puzzle threads, especially for n00bs like me!

skeptical scientist
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jordan wrote:Is there any chance that such a thing might be implemented for xkcd some time? It would be pretty useful in these puzzle threads, especially for n00bs like me!

This would be my preferred method as well. It's been suggested in the past, but so far nobody has done anything. I guess most people figure the system we have now works well enough that it's not worth messing with alternatives.
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

jordan
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skeptical scientist wrote:I guess most people figure the system we have now works well enough that it's not worth messing with alternatives.

I suppose that, given that xkcd readership will generally possess some level of visual acuity, it's unlikely that we will have any members who rely on screenreaders. And a working alternative has already been proposed anyway: separate threads for problems and solutions. So while it would be nifty, it's not really necessary.

Ansain
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Hey jordan If you look at my solution there is a different way to get the same answers out of them which will help you out in starting to think about the three gods. you will probably find this question easily yourself when you consider that the gods do not have ages.

pruwyben
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the answer i got involves some meta-questioning, so i don't know if it counts, but....

(spoiler)

the tricky thing about this phrasing is, the lying daughter would have to lie about lying, so she would give you the correct answer. to elaborate, if the answer to "is daughter #2 mischievous" were yes, the lying daughter would answer no, so if you ask her what her answer would be to that question, she would have to lie and say it would be yes. the truthful daughter would of course answer truthfully.

so if the answer is yes, either #1 or #2 is the mischievous daughter and you choose #3. if the answer is no, either #1 or #3 is mischievous, and you choose #2.

of course, this isn't exactly a yes or no question, since your not asking whether something is true, but rather what she would say, although the possible answers are still "yes" and "no". kinda weird...

Ansain
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that works but there is definately a good way to do this without meta-questioning

Buttons
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I'll admit to having not read all of the thread, but this doesn't seem too meta to me.
Do the statements "You are lying" and "You are the middle sister" have the same truth values?
The youngest and oldest should always answer "yes", and the middle should always answer "no". Is this the sort of thing we're going for? Alternately:
Do you have a sibling immediately older than you who is capable of telling statements with the same veracity as you are right now?
To which the middle child should always answer "yes" and the youngest and oldest should always answer "no".

I don't think we can get any less meta than that, since any question must reference whether the current speaker is telling the truth, or else we have the ambiguities mentioned on page one.

EDIT: Ah, I should have read the second page first. That one's much more clever.

Frankthetank
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or you could say, "Answer truthfully or I will kill you after the wedding. Are you the middle sister?"

forces all the sisters to tell the truth.

skeptical scientist
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Only if they believe you'll carry out the threat. Would you really kill your new bride after the wedding if she lied to you, knowing that you made the threat in public and that you will almost certainly be executed if you do?

By the way, if you intend to post on these forums more than once you should really post an intro post in the intro thread in general discussion. Posting in the intro thread is this board's turing test, and if you fail it, you may be treated as a bot and have your posts <strongbad>deleted</strongbad>.
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

DeeDee
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(Spoiler)
Ask a question that would force the liar and truth teller to say different answers (See many of the questions that could do this, some said in this thread.), then the middle sister will say the same as one of them. Pick one that did not answer the same as the other two. Remember that the thread creator said that "you want the liar or the truth teller" This method allows one to do such a thing, even without meta-questions.

"Focus on the solution, not the problem."

<< It's really simple.

Arkanin
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Deedee, I don't think we're allowed to ask every sister a question. Anyway I jumped the gun and said it wasn't solvable, but apparently there is a solution on page 2.

Ansain
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buttons your solution treats the middle sister as someone who is either a truthteller or a liar and has a random chance of each. however if she just answers each of your questions randomly then she could just as easily say no.

Buttons
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Ansain wrote:buttons your solution treats the middle sister as someone who is either a truthteller or a liar and has a random chance of each. however if she just answers each of your questions randomly then she could just as easily say no.

But she doesn't just answer questions randomly. Rather, for each question she chooses to either lie or tell the truth. There's a difference.

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