## Three Princesses with random middle sister

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

CortoPasta
Posts: 38
Joined: Fri Aug 15, 2008 5:51 pm UTC

### Three Princesses with random middle sister

Here is the original riddle:

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?

Spoiler:
Are you one of the youngest two daughters, or would you say that you are the middle daughter?

Oldest(Truth): No
//Not one of the youngest two daughters, nor would she say she is the middle daughter
Middle(While telling the truth): Yes
//Is one of the youngest two daughters, and would say she is the middle daughter
Middle(While telling a lie): Yes
/*Would say no to being one of the youngest two daughters, but the question "or would you say that you are the middle daughter?" Would be a double negative. {i.e. Because her answer would be no to "Are you the middle daughter?", she would lie and say "yes"{i.e. Because her answer(as a liar) would be no to "Are you the middle daughter?", she would lie and say "yes" to the part "would you say that you are the middle daughter?"}*/
Youngest(Liar): No
/*Would say no to being one of the youngest two daughters, but would have the opposite response to the double negative second part of the question and respond no {i.e. Because her answer(as a liar) would be yes to "Are you the middle daughter?", she would lie and say "no" to the part "would you say that you are the middle daughter?"} */

So using the answer above, pick the sister that says no.

And here is the challenge: If, instead of sometimes lying and sometimes telling the truth, the middle daughter's decision was made by a completely random coin toss, how do you tell her apart from the other two sisters?

Same rules apply (i.e. Only yes/no answers allowed. Not Yes/No/Don't know/Divide by zero/etc.), except you can ask as many questions you like. Who can get the highest probability of guessing correctly with the fewest questions asked?

ian
Posts: 706
Joined: Fri Mar 07, 2008 3:55 pm UTC
Location: Sealand

### Re: Three Princesses with random middle sister

edit: oops, didn't see the second bit, still might want to stick it in the other thread though

notzeb
Without Warning
Posts: 629
Joined: Thu Mar 08, 2007 5:44 am UTC
Location: a series of tubes

### Re: Three Princesses with random middle sister

I think you are assuming that the middle sister is less mischievous than she really is. She has been known to foil the most ridiculous logical paradoxen with a giggle and a grin.

Here's the real answer to the puzzle:
Spoiler:
Call the sisters A, B, and C. Ask sister A the following question:

"If I asked you, would you say that sister B is the middle sister?"

If she says yes, marry sister C. If she says no, marry sister B.
Anyways, isn't there already a thread with the same puzzle somewhere? And I think they must have the real solution somewhere within.
Zµ«V­jÕ«ZµjÖ­Zµ«VµjÕ­ZµkV­ZÕ«VµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­ZµkV­ZÕ«VµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­ZµkV­ZÕ«ZµjÖ­Zµ«V­jÕ«ZµjÖ­ZÕ«VµjÕ­Z

CortoPasta
Posts: 38
Joined: Fri Aug 15, 2008 5:51 pm UTC

### Re: Three Princesses with random middle sister

There is a thread for the original riddle, but I thought this took a different enough solution to warrant it's own. After nine pages it's hard to follow and I just thought this would be a fun challenge. I'll pull the thread if a few more people think it's redundant, just to save the mod's the time.

ian
Posts: 706
Joined: Fri Mar 07, 2008 3:55 pm UTC
Location: Sealand

### Re: Three Princesses with random middle sister

yes it's three down. just realised his extra challenge makes no difference at all,

Moonbeam
Posts: 292
Joined: Sat Dec 08, 2007 5:28 pm UTC
Location: UK

### Re: Three Princesses with random middle sister

So what exactly is the question ??

Are we simply trying to avoid marrying the middle sister (if so, it's been done before) ??

...... or do you want to be able to actually identify which one is the middle sister ??

Even though her choice is determined by the flip of a coin - she could (in theory) end up constantly giving the same reply as one of the other 2 sisters, thereby making it impossible to distinguish between the 2 .

If you're just after avoiding marrying the middle sister:

Spoiler:
You randomly choose one of the sisters and ask her the following question, whilst pointing to the other 2 sisters, "Is that princess younger than that princess". Whatever answer she gives - marry the one that she is making out to be the youngest of the 2.

Explanation: If you ask this question to the eldest princess (truth-teller), she will correctly identify who is the youngest out of the 2 remaining sisters (which happens to be the liar), so if you marry the sister that is indicated to be the youngest - you will end up marrying the liar in this case.

If you ask this question to the youngest sister (the liar), she will indicate that the truthful sister is the youngest out of the 2 (which is a lie - even though we don't know it). Marrying the sister that is indicated to be the youngest (the truth-teller in this case) will again ensure that we don't marry the middle sister, as we end up marrying the truthful sister in this case.

If you ask this question to the middle sister, it doesn't matter what answer she gives, as we're going to choose one of the other 2 sisters anyway. Choose the sister that she indicates to be the youngest (in this case you could end up marrying either of the other 2 sisters).

Avin
Posts: 91
Joined: Sun Nov 25, 2007 3:08 am UTC

### Re: Three Princesses with random middle sister

This question is much simpler. If the sister you pose the question to pulls out a coin and flips it before answering, don't marry her.

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

### Re: Three Princesses with random middle sister

notzeb wrote:Here's the real answer to the puzzle:
Spoiler:
Call the sisters A, B, and C. Ask sister A the following question:

"If I asked you, would you say that sister B is the middle sister?"

If she says yes, marry sister C. If she says no, marry sister B.
Anyways, isn't there already a thread with the same puzzle somewhere? And I think they must have the real solution somewhere within.

Except this puzzle seems to want us to identify the middle sister, while that puzzle asks us only to identify one person who we know is not the middle sister.

Anyways, the middle sister can be located
Spoiler:
In two questions, with 100% probability. First use the solution to the original puzzle to locate a sister who is not the middle sister. Then ask her, "If I were to ask you if she (point to another sister) is the middle sister, would you say 'yes'?" Either the eldest or the youngest sister will answer 'yes' if the middle sister is the one indicated, and otherwise 'no', so you can find the middle sister with certainty in two questions, one each to two different sisters. You cannot locate the middle sister with 100% probability in one question - for a proof, see the other thread - so this is optimal. I'm not sure what the best odds are of guessing the middle sister after asking only one question.
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

douglasm
Posts: 630
Joined: Mon Apr 21, 2008 4:53 am UTC

### Re: Three Princesses with random middle sister

This has been solved multiple times in the original puzzle's thread even with the original restriction of only getting a single question addressed to a single princess. I really don't think it needs its own thread.

jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

### Re: Three Princesses with random middle sister

Yeah, I'm locking. I'm not seeing the "identify the middle sister" thing that skep is (you definitely need two questions for that, or another possible answer from the sisters might do it, but I'm not seeing where that is asked), and otherwise its just the same problem as three princesses. The "the middle sister is random" interpretation is just the better way to interpret the three princesses problem.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.