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# 3 Princess Puzzle

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You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry.

The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time.

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

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

This is not a trick question, and the answer is not anything like "is this statement a lie?" The solution is nice and elegant.

Ask one princess, let's call her princess A, if princess B is older that princess C. You want to marry whichever one she indicates is younger.

--Youngest: she will tell you that the eldest one is younger

--Middle: either one is good

--Eldest: she will tell you that the youngest is younger

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Posted · Report post

There are 3 unknowns, and a Yes/No question discriminates between 2 and only 2 possibilities.

You don't need a complete solution.

Your choice either is or is not the mischievous one - two cases.

I believe there are at least three questions that work.

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Posted · Report post

The clunker in this riddle is Princess Mischievous.

Devise a strategy whereby after asking one of the Princesses a question

you will pick one of the other two as a wife. That way, if you happen

Either of the others is an acceptable choice!

All that's needed is a question which, if answered Yes [or No] by either

Princess Truth or Princess Lie, will identify Princess Mischievous.

A Yes/No question can guide a binary choice.

To optimize the strategy:

Since you will not choose her, ask the question of the ugliest of the three.

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Posted · Report post

I couldnt find a way around the Mischievious one answering either YES or NO randomly

Ask one princess, let's call her princess A, if princess B is older that princess C. You want to marry whichever one she indicates is younger.

--Youngest: she will tell you that the eldest one is younger

--Middle: either one is good

--Eldest: she will tell you that the youngest is younger

This wouldnt work. It has to be a YES or NO question, remember?

So say you do that, and line them up, 123. Ask Princess 1 if Princess 2 is older than Princess 3.

the truthteller will say the truth (YES or NO), the liar will lie, and the mischevious princess would also say either YES or NO.

Your own answer does not work. The middle one wouldnt say "either one is good", she would say a specific one is younger as well.

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Posted · Report post

The closest I've been able to get is asking #2 a certain q about #1, and if its NO then choose #3 and you're set, and if its YES then choose #1 and there's a 4/5 chance that you're okay. However this isnt the "perfect answer"

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Posted · Report post

This wouldnt work. It has to be a YES or NO question, remember?

"Is princess B older than princess C" is a yes or no question.

So say you do that, and line them up, 123. Ask Princess 1 if Princess 2 is older than Princess 3.

the truthteller will say the truth (YES or NO), the liar will lie, and the mischevious princess would also say either YES or NO.

The question the OP supplied works, it's just that the OP didn't explain why.

If you get a "yes" answer from Princess A, go with Princess C. If you get a "no" answer, go with Princess B.

Possible scenario:

Princess A: eldest-truthteller

Princess B: youngest- liar

Princees C: middle- normal

Princess A could only answer no, so by going with Princess B, you don't get the middle child.

Another scenario:

Princess A: youngest- liar

Princess B: eldest-truthteller

Princees C: middle- normal

Princess A could only answer no, so by going with Princess B, you don't get the middle child.

Another scenario:

Princess A: middle- normal

Princess B: eldest-truthteller

Princees C: youngest- liar

If Princess A answers yes, you pick C and get the youngest. If she answers no, you pick B and get the oldest. Either way you don't end up with her.

The middle one wouldnt say "either one is good", she would say a specific one is younger as well.

I think what the OP meant was that it doesn't matter if the middle child decides to tell the truth or lie, since you never pick the Princess you're asking the question to, you won't end up with her.

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Posted · Report post

This wouldnt work. It has to be a YES or NO question, remember?

"Is princess B older than princess C" is a yes or no question.

Yeah I know, as I said later

So say you do that, and line them up, 123. Ask Princess 1 if Princess 2 is older than Princess 3.

the truthteller will say the truth (YES or NO), the liar will lie, and the mischevious princess would also say either YES or NO.

The question the OP supplied works, it's just that the OP didn't explain why.

If you get a "yes" answer from Princess A, go with Princess C. If you get a "no" answer, go with Princess B.

Possible scenario:

Princess A: eldest-truthteller

Princess B: youngest- liar

Princees C: middle- normal

Princess A could only answer no, so by going with Princess B, you don't get the middle child.

Another scenario:

Princess A: youngest- liar

Princess B: eldest-truthteller

Princees C: middle- normal

Princess A could only answer no, so by going with Princess B, you don't get the middle child.

Another scenario:

Princess A: middle- normal

Princess B: eldest-truthteller

Princees C: youngest- liar

If Princess A answers yes, you pick C and get the youngest. If she answers no, you pick B and get the oldest. Either way you don't end up with her.

The middle one wouldnt say "either one is good", she would say a specific one is younger as well.

I think what the OP meant was that it doesn't matter if the middle child decides to tell the truth or lie, since you never pick the Princess you're asking the question to, you won't end up with her.

there are 6 possibilities, not 3

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Posted · Report post

Yeah I know, as I said later

Am I missing something? You said the OP's solution wouldn't work because it wasn't a yes or no question. I pointed out that the solution given was a yes or no question. I don't see where you later retracted that comment.

there are 6 possibilities, not 3

Would you like me to go over all possible scenarios to show you that they all work? I will if you like, but I was pretty confident that you could do that on your own after I pointed out how the solution works.

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Posted · Report post

I believe there are at least three questions that work.
Here are four questions that work.

Select the least desirable Princess and call her A.

[1] is B older than C?

[2] does B lie less frequently than C?

[3] is B more likely to lie than you?

[4] if I were to ask, would you say that B is Princess Mischievous?

In each case, if the answer is Yes choose C; if No, choose B.

Princess Truth and Princess Lie will answer Yes if Princess C is the other of these two, and No if Princess B is.

If A is Princess Mischievous, a random B or C choice gets you a winner either way.

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Posted · Report post

Yeah I know, as I said later

Am I missing something? You said the OP's solution wouldn't work because it wasn't a yes or no question. I pointed out that the solution given was a yes or no question. I don't see where you later retracted that comment.

I was just pointing out that he didnt give his answer in YES or NO form. If you look at the next line I said:

So say you do that, and line them up, 123. Ask Princess 1 if Princess 2 is older than Princess 3.
which is a YES or NO question, I assumed that was the YES or NO question the OP asked with those answers. Nvm. It doesnt matter
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Posted · Report post

Here are four questions that work.

Select the least desirable Princess and call her A.

[1] is B older than C?

[2] does B lie less frequently than C?

[3] is B more likely to lie than you?

[4] if I were to ask, would you say that B is Princess Mischievous?

In each case, if the answer is Yes choose C; if No, choose B.

Princess Truth and Princess Lie will answer Yes if Princess C is the other of these two, and No if Princess B is.

If A is Princess Mischievous, a random B or C choice gets you a winner either way.

I see how 1 and 2 work (because they are basically the same as OP's answers.),

but I don't think 3 or 4 work.

3)

Okay, so if I ask Honest (A) if B is more likely to lie than her, she says yes. But, she says yes whether B is Mischievous or Liar, so I may end up with Mischievous, never mind the fact that I don't know that A is honest.

The case is the same with Liar. She will say yes to either one so there is no way to distinguish between them.

So basically, if you ask the question of Liar or Honest, you have a 50% chance.

4)

Okay, so I ask A if B is Mischievous. She says yes. I don't know whether A is Honest or Liar (or Misch but that point is moot since you're a winner either way if you ask her), so do I pick B or C?

So, again, my future lies on chance. Not cool.

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Posted · Report post

I don't think 3 or 4 work.

3) Okay, so if I ask Honest (A) if B is more likely to lie than her, she says yes.

But, she says yes whether B is Mischievous or Liar, so I may end up with Mischievous,

never mind the fact that I don't know that A is honest.

The case is the same with Liar.

She will say yes to either one so there is no way to distinguish between them.

So basically, if you ask the question of Liar or Honest, you have a 50% chance.

You're right - [3] is a broken version of [2].

I had something else in mind. Good catch..

4) Okay, so I ask A if B is Mischievous. She says yes.

I don't know whether A is Honest or Liar (or Misch but that point is moot since you're a winner either way if you ask her), so do I pick B or C?

So, again, my future lies on chance. Not cool.

Actually, [4] works. Remember, I said ...

In each case, if the answer is Yes choose C; if No, choose B.

PT and PL will both reliably identify PM.

If A is PT, she tells the truth about telling the truth, and if A ls PL, she lies about lying.

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