## Three princesses

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

torne
Posts: 98
Joined: Wed Nov 01, 2006 11:58 am UTC
Location: London, UK

Are there three princesses in it?

jpark
Posts: 6
Joined: Fri Jun 15, 2007 3:29 pm UTC
Try this:

The question is: If I were to ask you if you were the eldest sister, what would you answer?

If you are asking the eldest sister, she will say "Yes".

If you are asking the younger sister, she would have said yes to the direct question, but will lie and say "No".

If you are asking the middle sister and she has chosen to lie, her answer will model the younger sister and she will say "No". If she has chosen to tell the truth, she will say "No".

skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco
jpark - you may only ask one of the sisters the question, so if you get an answer of "no", you don't know who (not) to marry.
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

jpark
Posts: 6
Joined: Fri Jun 15, 2007 3:29 pm UTC
My question presumed you wanted to marry the eldest sister. It could be rewritten if you wanted to marry the youngest.

jpark
Posts: 6
Joined: Fri Jun 15, 2007 3:29 pm UTC
You are correct. If I get a "No', I still don't have an answer.

lamexorleet
Posts: 1
Joined: Fri Jun 15, 2007 3:58 pm UTC

### perhaps some can reduce this further ?

My Question would be :
"
Do you sometimes tell the truth XOR is it correct than neither of your sisters always tell the truth ?"

If only the suitibility of all offerant partners could be decided by boolean logic.

Will she be my eXclusive OR tonite ?

anon
Posts: 1
Joined: Fri Jun 15, 2007 4:10 pm UTC

### Not asking the right question.

Nice puzzle.

Most people don't seem to be asking the right questions about the question you need. You don't care about finding out who you are talking to. You just want to avoid marrying the middle sister.

So, to all you naysayers out there, there is a single-well defined question that will, if used correctly, ensure that you do not marry the middle sister.

amylynne321
Posts: 1
Joined: Fri Jun 15, 2007 3:55 pm UTC

### Line Them Up!!

The question has to be something like:

"If I were to ask her (point to a different sister) if she always lies, would she say yes."

Posts: 1
Joined: Fri Jun 15, 2007 4:24 pm UTC
Contact:

I saw this puzzle and realized it was just a derivative of the following logic problem presented here: http://en.wikipedia.org/wiki/The_hardes ... uzzle_ever

Be warned, possible answers are presented near the bottom. I added more solutions about 5 days ago. Enjoy!

skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

### Re: perhaps some can reduce this further ?

lamexorleet wrote:My Question would be :
"
Do you sometimes tell the truth XOR is it correct than neither of your sisters always tell the truth ?"

If only the suitibility of all offerant partners could be decided by boolean logic.

Will she be my eXclusive OR tonite ?

The middle sister might also answer no, so this is not 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.

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

juhraffe
Posts: 1
Joined: Fri Jun 15, 2007 4:47 pm UTC

### solution?

Ask daughter A, "If I were to ask you if daughter B is the middle daughter, would you answer 'YES'?"

If YES, choose daughter C
If NO, choose daughter B

skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco
That works.
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

jpark
Posts: 6
Joined: Fri Jun 15, 2007 3:29 pm UTC
Let me try again with the logic I applied incorrectly in my former post.

The question is: If I were to ask if you were the middle daughter, what would you say?

The elder daughter will answer "No".
The younger daughter would have said yes, but lies and says "No".
The middle daughter, if she chooses to tell the truth, will say "Yes". If she chooses to lie, she would have said "No", but lies and says "Yes"

If the daughter you ask the question says "No", marry her. If she says "Yes", marry either of the remaining daughters.

buckX
Posts: 1
Joined: Fri Jun 15, 2007 6:12 pm UTC
Couldn't you just ask each sister, are you the middle sister?
Oldest says no, youngest says yes, middle says either yes are no. If two say yes and one says no, pick the one that says no, if two say no and one says yes, pick the one that says yes.

jpark
Posts: 6
Joined: Fri Jun 15, 2007 3:29 pm UTC
You are only allowed one question to one daughter.

skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco
jpark, she might decide that she'll tell the truth, but she would have lied had you asked her if she was the middle daugher, and therefore say "no" as well. You can't assume she'll choose the same truth value for her answer to your actual question as she chooses in the hypothetical situatation she's considering.
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

rippond
Posts: 2
Joined: Fri Jun 15, 2007 7:03 pm UTC

We know that the youngest one always lies, the oldest always tells the truth, and the other is usually lies but sometimes tells the truth.

The solution lies with a question about the age of the other sisters.

Are both your sisters younger than you?

If it is the truthful eldest, the answer is 'Yes.'
If it is the middle, the answer could be either.

If the princess answers yes, there is a good chance it is either the youngest or oldest....

The rules are that you may only ask one question to one princess, so I suppose you could ask the same question to the same princess many times to see if it is the middle princess if she changes her answer or not.

What an interesting problem.

jpark
Posts: 6
Joined: Fri Jun 15, 2007 3:29 pm UTC
skeptical scientist,

I thought the puzzle stated the middle sister made a choice to tell the truth or to lie. This is an either/or situation. The situation you postulate is a mixture of truth and falsehood.

If we allow the middle sister to do both, we diverge from the rules of the puzzle.

Albion Tourgee
Posts: 1
Joined: Fri Jun 15, 2007 7:49 pm UTC

### Another approach

What about, are you the one I'm going to marry? The question can be answered, but only by the middle sister. Perhaps not as elegant as the first solution, but shorter.

milkman dan
Posts: 1
Joined: Fri Jun 15, 2007 7:54 pm UTC

### One princess doesn't get stuck

I will give you a hint: you can ask a paradoxical question that the princesses that /always/ tell the truth cannot answer. The inconsistent sister will be able to answer it. So you ask the question and if you get any answer you pass. If you get an answer you ask her to marry you.

Remarc
Posts: 1
Joined: Fri Jun 15, 2007 7:57 pm UTC
Location: Florida
Contact:
I don't know if somebody already posted a question like this...
But couldn't you just ask: "Is the king your father?" or "How many questions can I ask you?"
Since you already know the answer to those questions, whichever sister(s) answer(s) correctly is going to be the eldest and middle sisters...?

rufus
Posts: 22
Joined: Wed May 30, 2007 2:02 pm UTC
Location: Pittsburgh
This logic puzzle is fatally flawed in that I would much rather have the mischevious (sp?) middle sister.
Human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so.

gmalivuk
GNU Terry Pratchett
Posts: 26829
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:
You have to find out which is the oldest or the youngest. Knowing the one you asked is the oldest or middle doesn't help you.

And to skeptical scientist, I would assume that the middle sister must decide beforehand whether she will lie or tell the truth. Otherwise, there isn't a solution. (It's not her answer that's random, but whether she tells the truth. These are different cases.)
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

sulayman
Posts: 3
Joined: Fri Jun 15, 2007 8:35 pm UTC
Pick a sister and ask her what 1 + 1 equals.

If she answers "2," marry her. At the very least you'll wind up with a wife that lies half the time, which is still better than most women.

jjoseph_x
Posts: 4
Joined: Fri Jun 15, 2007 8:40 pm UTC
If the middle sister can either lie or tell the truth, then the youngest and eldest sisters wouldn't be able to tell you what her answer would be: they wouldn't know.

So if you asked any one of a question like "would the middle sister tell me the truth" the oldest and youngest would have to answer: "I don't know". Only the middle sister would be able to give a yes or no answer, depending on whether she's lying.

sulayman
Posts: 3
Joined: Fri Jun 15, 2007 8:35 pm UTC
Not necessarily. The middle sister could also answer "I don't know" in reference to herself. It would work as either an honest or dishonest answer.

Posts: 3
Joined: Fri Jun 15, 2007 8:47 pm UTC
It's simple...

Ask sister 1 if sister 2 lies more than sister 3.
If answer is yes: pick sister 2
If answer is no: pick sister 3

terr
Posts: 1
Joined: Fri Jun 15, 2007 8:43 pm UTC

### my two cents

I agree with Albion Tourgee. He was the first to point out a question that I was thinking about. jjoseph_x also said it. The question must be only answerable to someone that will shift their answer to give you one. The other two wouldn't know even if the lier would lie.

So you ask the question and if the girl says "I don't know" you marry her (since you will marry the lier). If she answers a "Yes" or "No" you pick one of the others.

jjoseph_x
Posts: 4
Joined: Fri Jun 15, 2007 8:40 pm UTC
sulayman wrote:Not necessarily. The middle sister could also answer "I don't know" in reference to herself. It would work as either an honest or dishonest answer.

If the middle sister or the youngest sister can answer "I don't know" (especially as a way of lying) then it'd be impossible to answer the question... there'd be too many variables, since you can only ask one sister one question.

John Stossel
Posts: 1
Joined: Fri Jun 15, 2007 8:18 pm UTC

### Visual Solution

Here is the solution I came up with, tried to present it in rows hope the formatting comes out correctly.
I would line the three daughters in a row and ask the first girl if the second daughter is older than the third.
I have labelled the daughters A=Eldest B=Middle C=Youngest
The permutations are (from top to bottom)
1.A = Y
2.B If Y, Choose 3 = C Youngest
3.C

A = N
C If N, Choose 2 = C Youngest
B

B = Y or N
A Irrelevant but follow rules above will end up with A or C.
C

B = Y or N
C Irrelevant but follow rules above.
A

C = N
A If N, Choose 2 = A Oldest
B

C = Y
B If Y, Choose 3 = A Oldest
A

So with one question by following above rules you should alway end up with the eldest or the youngest. Hope this helps, let me know if it is correct.

sulayman
Posts: 3
Joined: Fri Jun 15, 2007 8:35 pm UTC

Ask sister 1 if sister 2 lies more than sister 3.
If answer is yes: pick sister 2
If answer is no: pick sister 3

I'm lost. If you don't know which of the sisters lies, truths, and flipflops, how can you know that "Sister 1" isn't a viable option.

What if Sister 1 is the Honest Sister? She'd answer "Yes" if Sister 2 were the Liar, and "No" if Sister 2 were the flipflopper. And in either case, you'd be picking the wrong sister.

Posts: 3
Joined: Fri Jun 15, 2007 8:47 pm UTC
sulayman wrote:

Ask sister 1 if sister 2 lies more than sister 3.
If answer is yes: pick sister 2
If answer is no: pick sister 3

I'm lost. If you don't know which of the sisters lies, truths, and flipflops, how can you know that "Sister 1" isn't a viable option.

What if Sister 1 is the Honest Sister? She'd answer "Yes" if Sister 2 were the Liar, and "No" if Sister 2 were the flipflopper. And in either case, you'd be picking the wrong sister.

You don't need to pick the honest sister, you just need to not pick the mischevious/random one.

jjoseph_x
Posts: 4
Joined: Fri Jun 15, 2007 8:40 pm UTC

Ask sister 1 if sister 2 lies more than sister 3.
If answer is yes: pick sister 2
If answer is no: pick sister 3

The problem is that you don't know which one is sister 3 or sister 2.

Let's say that sister 3 is the middle one and that sister 2 is the eldest. If you ask the lier your question, she'll answer "no" (since the middle does lie more, and the lier will...well... lie), so you pick sister 3 (the one that you wanted to avoid).

skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco
jpark wrote:skeptical scientist,

I thought the puzzle stated the middle sister made a choice to tell the truth or to lie. This is an either/or situation. The situation you postulate is a mixture of truth and falsehood.

If we allow the middle sister to do both, we diverge from the rules of the puzzle.

The middle sister is not doing both. She is truthfully saying "no", meaning "if you had asked me if I were the middle sister, I would have lied and said I wasn't". Since you're asking what she would have said had you asked a direct question, she can truthfully tell you that she would have lied without contradicting anything in the statement of the riddle.

gmalivuk wrote:You have to find out which is the oldest or the youngest. Knowing the one you asked is the oldest or middle doesn't help you.

You don't need to figure out the identity of any of the sisters. You merely need to find a strategy which guarantees that you not marry the middle.

gmalivuk wrote:And to skeptical scientist, I would assume that the middle sister must decide beforehand whether she will lie or tell the truth. (It's not her answer that's random, but whether she tells the truth. These are different cases.)

Assuming facts not in evidence in a logic puzzle is a sure route to trouble. There is nothing in the statement of the puzzle to suggest that this is the case. And I'm not disagreeing that her decision to tell the truth, not her answer, is random. I'm just saying that if the question is about a hypothetical, there is no reason to believe that the middle sister's actual answer will have the same truth value as the hypothetical middle sister's hypothetical answer.

gmalivuk wrote:Otherwise, there isn't a solution.

This is simply false. Multiple correct answers can be found in this thread, which all exploit the same basic principle which all correct answers must share.
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

imonsei
Posts: 5
Joined: Fri Jun 15, 2007 10:15 pm UTC
if they worship a diety that is all powerfull
ask the one "can <diety> make a stone so big <diety> cannot lift it?"
the truth telling one cannot answer this question at all, since a yes or a no will go against her faith and produce her to believe she is telling a lie.
if you ask the flipflopper she will answer yes or no since she doesnt care.
the liar will answer yes or no, because nomatter what she answers she will believe it to be a lie.

there now you don't get to marry the liar or the flipflopper, so bonus points for my solution.

now as an aside i ask you. how can <diety> produce that stone and still continue to be <diety>? (there is an answer)
Last edited by imonsei on Fri Jun 15, 2007 10:41 pm UTC, edited 1 time in total.

funkyllama
Posts: 2
Joined: Fri Jun 15, 2007 9:59 pm UTC

Hello,

I'm really excited because I'm terrible at these (despite having studied a lot of maths) but I'm pretty sure that this time I got the (an) answer... I'll try to give a useful hint, in white text as others have done...

It seems many people are worrying along these lines: "the middle sister could answer yes or no to anything, how do you distinguish the meaning of her answers from those of the other sisters?"

That's a big stumbling block, but here's a three-step hint on how to overcome it. I think it still leaves a lot of work to get the solution. However, maybe you shouldn't look at step 3 unless you've almost given up:

1. As many have already noted, you can't distinguish the meaning of her answer.

2. In other words, you may have chosen the eldest, youngest, or middle sister, and if it was the middle sister, you have no way of interpreting what her answer means.

3. So to deal with this case: Whoever you ask, whatever your question is, and whatever the answer, you will always have to somehow choose one of the OTHER two sisters to marry... after all, if you do get the middle sister, her answer doesn't matter: either of her sisters is good!

Whether you use this or not, I believe you can get to the answer without mystical means (or death-defying logical contortions)...

Posts: 3
Joined: Fri Jun 15, 2007 8:47 pm UTC
jjoseph_x wrote:

Ask sister 1 if sister 2 lies more than sister 3.
If answer is yes: pick sister 2
If answer is no: pick sister 3

The problem is that you don't know which one is sister 3 or sister 2.

Let's say that sister 3 is the middle one and that sister 2 is the eldest. If you ask the lier your question, she'll answer "no" (since the middle does lie more, and the lier will...well... lie), so you pick sister 3 (the one that you wanted to avoid).

Sister 1 = youngest, Liar
Sister 2 = eldest, Truthful
Sister 3 = middle, Random

So my answer says "ask sister 1 (liar) if sister 2 (truthful) lies more than sister 3 (random)". Since sister 1 is a liar, she answers falsly "Yes, sister 2 (truthful) lies more than sister 3 (random)", not "No" as you had her reply in your post.

funkyllama
Posts: 2
Joined: Fri Jun 15, 2007 9:59 pm UTC

Oops, somehow I didn't see all the posts giving the answer. Darn multi-page threads. No point in "hinting" then!

I believe karmadillo is correct (I arrived at a similar answer with the opposite question: "does sister 2 tell the truth more than sister 3?" or equivalently "is sister 2 older than sister 3?")

If you list all the possible arrangements of the sisters, and all the possible answers, you should find that you always end up with the liar or the honest one - but never the middle sister.

jjoseph_x
Posts: 4
Joined: Fri Jun 15, 2007 8:40 pm UTC
jjoseph_x wrote:

Ask sister 1 if sister 2 lies more than sister 3.
If answer is yes: pick sister 2
If answer is no: pick sister 3

The problem is that you don't know which one is sister 3 or sister 2.

Let's say that sister 3 is the middle one and that sister 2 is the eldest. If you ask the lier your question, she'll answer "no" (since the middle does lie more, and the lier will...well... lie), so you pick sister 3 (the one that you wanted to avoid).

Sister 1 = youngest, Liar
Sister 2 = eldest, Truthful
Sister 3 = middle, Random

So my answer says "ask sister 1 (liar) if sister 2 (truthful) lies more than sister 3 (random)". Since sister 1 is a liar, she answers falsly "Yes, sister 2 (truthful) lies more than sister 3 (random)", not "No" as you had her reply in your post.

Ooops. You're absolutely right.

ggrot
Posts: 3
Joined: Sat Jun 16, 2007 12:23 am UTC
Ask Sister A the following question:

"What would the sister whose age is most different from yours say to the question: 'Is Sister B the middle-aged sister?' ?"

Irregardless of the answer, you don't pick sister A - so if sister A is the middle sister, it doesn't matter what her answer is, you win.

That leaves two possibilities: Sister A is the truth teller or Sister A is the liar. Either way, the answer will be the same.

If the answer is Yes, marry Sister C. If the answer is No, marry sister B.