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#1
11-19-2009, 07:16 PM
 HoboStew Guest Join Date: Nov 2006 Posts: 1,158
A twist on the old one liar on truth teller riddle

There is a king. Not a crazy, murderous king like the other riddles, this one is actually kind and wealthy, and as such you would like to marry one of his three daughters. These daughters, while lovely creatures, can only answer yes or no questions. The eldest daughter always tells the truth, the middle daughter always lies. The third daughter is crazy and will answer any question randomly, either yes or no. You crave stability in life. You would like to marry the one who always tells the truth, but you are willing to settle for the one who lies all the time just because at least you will know where you stand. You do NOT want to marry the crazy one under any circumstances.

#2
11-19-2009, 07:18 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,526
We did essentially the same problem very recently.

SPOILER:
Any question you want to ask, you can easily reword into such a fashion as that you'll get a straight answer from anyone except the crazy one; instead of asking "Is P true?", ask "Is it the case that either you're the truth-teller and P is true, or you're the liar and P is false?". You'll get the right answer to "Is P true?" from either the truth-teller or the liar, and you'll get a random answer from the crazy one.

Using that technique, go up to Person A and ask "Is Person B the crazy one?" (i.e., actually ask "Is it the case that either you're the truth-teller and Person B is the crazy one, or you're the liar and Person B is not the crazy one?"). If the answer you get is "Yes", then you know either A or B is crazy, and you can marry C. Otherwise, either A or C is crazy and you can marry B.

Last edited by Indistinguishable; 11-19-2009 at 07:22 PM.
#3
11-19-2009, 07:23 PM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 78,093
I don't feel like going through the gymnastics of actually constructing the questions, but first, we need to use one of the standard tricks to get good information out of a liar, common in puzzles of this sort. Then, we need
SPOILER:
to ask one of the girls which of her sisters I should marry. If the one I ask is the truth-teller, then she'll answer honestly, and I'll get the liar (whom I can settle for). If the one I ask is the liar, then I'll still get good information out of her, and end up marrying the truth-teller. And if the one I ask is the crazy daughter, then I don't know if I'll get the liar or the truth teller, but since I'm taking one of the questioned girl's sisters, I still won't end up with the crazy one.

I presume that there will be time after the wedding to figure out whether the lass I got is the liar or truth-teller, but that can't be determined from the single question.
#4
11-19-2009, 07:35 PM
 HoboStew Guest Join Date: Nov 2006 Posts: 1,158
Chronos has it. You don't need the compound questions to solve this one (and had I thought about it I would have explicitly forbid them anyways since they are cheating )

Took me an hour to figure this one out. The correctly phrased question is
SPOILER:
Assuming you ask person A, "Does person B lie more than person C?" If they answer yes, marry that one else the other one.
#5
11-19-2009, 07:43 PM
 RandMcnally Guest Join Date: Apr 2003 Location: Santa Barbara, CA Posts: 7,740
#6
11-20-2009, 12:30 AM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 78,093
Oh, that one's easy. Anyone who follows XKCD knows that you can't trust the guy in the hat.
#7
11-20-2009, 12:36 AM
 DanBlather Guest Join Date: Jul 2001 Location: Portland, O-bamagon Posts: 7,851
You just marry the one with the biggest tits.
#8
11-20-2009, 06:37 AM
 twickster Charter Member Charter Member Join Date: Aug 2002 Location: Philadelphia Posts: 38,721
This doesn't seem right for Cafe Society, so I'll move it over to the Game Room.

twickster, Cafe Society moderator
#9
11-20-2009, 07:01 AM
 Gyrate Guest Join Date: Nov 2000 Location: Deepest South London Posts: 21,382
Quote:
 Originally Posted by RandMcnally XKCD's take on this riddle.
And the Order of the Stick take the Gordian Knot approach.
#10
11-20-2009, 09:25 AM
 Munch Guest Join Date: Mar 2000 Location: Parts Unknown Posts: 22,084
Quote:
 Originally Posted by Chronos SPOILER: to ask one of the girls which of her sisters I should marry. If the one I ask is the truth-teller, then she'll answer honestly, and I'll get the liar (whom I can settle for). If the one I ask is the liar, then I'll still get good information out of her, and end up marrying the truth-teller. And if the one I ask is the crazy daughter, then I don't know if I'll get the liar or the truth teller, but since I'm taking one of the questioned girl's sisters, I still won't end up with the crazy one.
Does that work? The answers I'd expect from that question ("which of your sisters should I marry?"):
Truth: Liar
Liar: Crazy
Crazy: Truth or Liar

Crazy's still an option. Hobo's question eliminates that.
#11
11-20-2009, 10:31 AM
 glee Guest Join Date: Aug 1999 Location: Obama country Posts: 15,115
In this case, you want to reach the native village.
You arrive at a fork in the road, where there are three natives gathered : one truthteller, one liar and one answers at random.

SPOILER:
Did you know they are serving free beer in the village?

Then you simply
SPOILER:
#12
11-20-2009, 11:22 AM
 HoboStew Guest Join Date: Nov 2006 Posts: 1,158
Quote:
 Originally Posted by Munch Does that work? The answers I'd expect from that question ("which of your sisters should I marry?"): Truth: Liar Liar: Crazy Crazy: Truth or Liar Crazy's still an option. Hobo's question eliminates that.
Keep in mind you can only answer yes or no questions, Chronos was using his question as shorthand for mine, but they are essentially the same question.
#13
11-20-2009, 11:59 AM
 Really Not All That Bright Guest Join Date: May 2003 Location: Florida Posts: 67,707
Quote:
 Originally Posted by DanBlather You just marry the one with the biggest tits.
Winner.
#14
11-20-2009, 12:01 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,526
Quote:
 Originally Posted by HoboStew Chronos has it. You don't need the compound questions to solve this one (and had I thought about it I would have explicitly forbid them anyways since they are cheating )
Well, there's no use outlawing them, if you're allowed to ask any questions about the role-assignments at all, since they amount to the same thing. After all, any such question amounts in the end to "Is it the case that the role assignments are like this or this or ...?", where the various "this"es are the role assignments on which the answer is "Yes". Which is to say, "Is it the case that (you're a truth-teller and everyone else's roles are like this or this or ...) or (you're a liar and everyone else's roles are like this or ...) or (you're crazy and everyone else's roles are like this or ...)?", just organizing the assignments like so. Phrasing a question as a "compound question" is just making this explicit.

For example, with yours:
SPOILER:
"Does B lie more than C?" amounts to "Is it the case that either B is the liar or C is the truth-teller?" which amounts to "Is it the case that either (I'm not the liar and B is) or (I'm not the truth-teller and C is)?" which amounts to "Is it the case that either ((I'm the truth-teller or crazy) and B is the liar) or ((I'm the liar or crazy) and C is the truth-teller)?" which amounts to "Is it the case that either (I'm the truth-teller and B is the liar) or (I'm the liar and C is the truth-teller) or (I'm crazy and either B is the liar or C is the truth-teller)?".

And, of course, the final clause about what the right answer is when A is crazy can be modified willy-nilly without changing what kind of responses you might get, since, when A is crazy, she'll just ignore your question and give a random response anyway. So getting rid of it for simplification, you end up with "Is it the case that either (I'm the truth-teller and B is the liar (and thus C is the crazy one)) or (I'm the liar and C is the truth-teller (and thus B is the crazy one))?". Which [swapping B for C] was my question. [In fact, every question that works is isomorphic to my question in this sense]

Last edited by Indistinguishable; 11-20-2009 at 12:03 PM.
#15
11-20-2009, 12:21 PM
 Chronos Charter Member Moderator Join Date: Jan 2000 Location: The Land of Cleves Posts: 78,093
Right, the actual question would have been something like "If I were to ask you if sister B was consistent, what would you say?".

HoboStew's question is a bit more elegant, but it runs the risk that they're not all equally talkative: The crazy lady might lie more than the actual liar, if she talks twice as much.
#16
11-20-2009, 04:40 PM
 Death of Rats Guest Join Date: Aug 2007 Location: The Barack of Obama Posts: 3,548
Quote:
 Originally Posted by DanBlather You just marry the one with the biggest tits.
What if the liar is wearing falsies?
#17
11-21-2009, 01:14 PM
 SCSimmons Guest Join Date: Mar 2001 Location: Arlington, TX Posts: 3,264
Quote:
 Originally Posted by HoboStew You do NOT want to marry the crazy one under any circumstances.
I do not have any new contribution to offer to the logical puzzle, which seems to me to have been handled admirably. I would like to commend to your attention the quoted sentence of the puzzle, which seem to me to be the best advice of any kind that I have ever seen.
__________________
-Christian
"You won't like me when I'm angry. Because I always back up my rage with facts and documented sources." -- The Credible Hulk
#18
11-21-2009, 02:44 PM
 Xema Guest Join Date: Mar 2002 Posts: 11,791
Quote:
 Originally Posted by SCSimmons ... the best advice of any kind that I have ever seen.
But often hard to put into practice. Those who always tell the truth or always lie typically are thin on the ground, whereas crazy seems to be eternally popular.
#19
11-21-2009, 04:29 PM
 Carmady Guest Join Date: Aug 2008 Posts: 1,474

The crazy daughter answers randomly, therefore you cannot be sure that she has not lied more than the liar. For that matter, you cannot be sure that she hasn't been consistent so far in her life by pure coincidence. Maybe she rarely speaks, which makes sense given her limited vocabulary.

Further, it is questionable to assume the daughters know that they fall into the three categories, or who falls into which. The king knows, but maybe they don't.

The XKCD forum has a working answer, which instead relies on everyone being aware of the relative ages of the daughters.

I would prefer an answer that does not rely on that though. What if absolutely all you know is that one daughter always lies, one always tells the truth, and one is crazy? Nothing about ages, nothing about the daughters knowing who is what.

Last edited by Carmady; 11-21-2009 at 04:31 PM.
#20
11-21-2009, 04:41 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,526
Quote:
 Originally Posted by Carmady Hobostew's answer does not work. The crazy daughter answers randomly, therefore you cannot be sure that she has not lied more than the liar. For that matter, you cannot be sure that she hasn't been consistent so far in her life by pure coincidence. Maybe she rarely speaks, which makes sense given her limited vocabulary.
Just rephrase Hobostew's answer then; reword the binary relation "x lies more than y" as "either x is the liar or y is the truth-teller", for example. As I said, this is the unique solution, up to isomorphism.

Quote:
 Further, it is questionable to assume the daughters know that they fall into the three categories, or who falls into which. The king knows, but maybe they don't. The XKCD forum has a working answer, which instead relies on everyone being aware of the relative ages of the daughters.
I can't see how this would be possible (see below). But perhaps I misunderstand what you are stating. What's the answer the xkcd forum gives?

Quote:
 I would prefer an answer that does not rely on that though. What if absolutely all you know is that one daughter always lies, one always tells the truth, and one is crazy? Nothing about ages, nothing about the daughters knowing who is what.
Then you can't do it. You put your one question to one of the daughters (let's call her A) and she gives you information which is supposed to guarantee you the ability to figure out a non-crazy one. No matter what she says, it would be compatible with the possibility that she herself is crazy and just answering randomly; thus, since you cannot hope to rule her craziness out, one of her answers has to indicate that B is non-crazy and the other has to indicate that C is non-crazy. But if she lacks any information about which is which, she cannot modify her behavior accordingly; e.g., she would have to give the same answer whether she was True and B was crazy or she was True and C was crazy.

Last edited by Indistinguishable; 11-21-2009 at 04:46 PM.
#21
11-21-2009, 04:59 PM
 Carmady Guest Join Date: Aug 2008 Posts: 1,474
Quote:
 Originally Posted by Indistinguishable Just rephrase Hobostew's answer then; reword the binary relation "x lies more than y" as "either x is the liar or y is the truth-teller", for example.
True.

Quote:
 Originally Posted by Indistinguishable I can't see how this would be possible (see below). But perhaps I misunderstand what you are stating. What's the answer the xkcd forum gives?
In their statement, the youngest daughter always lied, middle was crazy, and eldest tells the truth. It can easily be converted to this scenario.

The solution is to ask A "Is B younger than C?"

Which of course is the exact same question as yours, but does not rely on the sisters knowing which category each falls into.

Quote:
 Originally Posted by Indistinguishable Then you can't do it. You put your one question to one of the daughters (let's call her A) and she gives you information which is supposed to guarantee you the ability to figure out a non-crazy one. No matter what she says, it would be compatible with the possibility that she herself is crazy and just answering randomly

You don't need an answer from the crazy daughter at all, if you never choose the daughter that you ask a question to.

You only have to get a meaningful answer from both the liar and truth teller.

But you are right, I'm pretty sure it is impossible without some extra information.

Last edited by Carmady; 11-21-2009 at 05:02 PM.
#22
11-21-2009, 05:04 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,526
Quote:
 Originally Posted by Carmady In their statement, the youngest daughter always lied, middle was crazy, and eldest tells the truth. It can easily be converted to this scenario. The solution is to ask A "Is B younger than C?" Which of course is the exact same question as yours, but does not rely on the sisters knowing which category each falls into.
Huh? How does it not require that? A needs to know their relative ages in order to answer that question, and the relative ages are stipulated to correspond to the answering-formats, so A knows that as well.

Quote:
 You don't need an answer from the crazy daughter at all, if you never choose the daughter that you ask a question to. You only have to get a meaningful answer from both the liar and truth teller.
The point is, if you only get an answer from one daughter, then you can't rule out the possibility that that daughter is crazy. So you need to set things up so that, no matter who your question goes to, the two possible responses each indicate one of the other two daughters as non-crazy.

Last edited by Indistinguishable; 11-21-2009 at 05:08 PM.
#23
11-21-2009, 05:09 PM
 Carmady Guest Join Date: Aug 2008 Posts: 1,474
Quote:
 Originally Posted by Indistinguishable Huh? How does it not require that? A needs to know their relative ages in order to answer that question, and the relative ages are stipulated to correspond to the answering-formats, so A knows that as well.

I don't follow. Just because the daughters know their relative ages doesn't mean they know anything about their relative attitudes with regard to the truth.

The problem stipulates that you know how age corresponds to truth telling, but they might not.
#24
11-21-2009, 05:12 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,526
Quote:
 Originally Posted by Carmady I don't follow. Just because the daughters know their relative ages doesn't mean they know anything about their relative attitudes with regard to the truth. The problem stipulates that you know how age corresponds to truth telling, but they might not.
Well, alright, but that's just a trivial relabelling of the problem as it exists. I could further provide a solution where the daughters needn't know their truth-attitudes nor their relative ages, just what they had for breakfast that morning. The stipulation is provided that the one which ate oatmeal is the youngest and a liar, the one who ate pancakes is the middle child and crazy, and the one who ate waffles is the oldest and honest, though the daughters may not know this. But nothing is actually gained by this solution, and nothing's really been changed about the problem; I've just reworded "liar" into "oatmeal-eater" and so on.

Last edited by Indistinguishable; 11-21-2009 at 05:13 PM.
#25
11-21-2009, 05:15 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,526
Quote:
 Originally Posted by Indistinguishable The point is, if you only get an answer from one daughter, then you can't rule out the possibility that that daughter is crazy. So you need to set things up so that, no matter who your question goes to, the two possible responses each indicate one of the other two daughters as non-crazy.
Which, as pointed out above, means there's essentially only one working solution: "You there! If you're honest, point to which of your two sisters I should marry. If you're a liar, point to which of your two sisters I shouldn't marry [of course, you being a liar, you'll do the opposite of what I say, and point to the one I should marry]. And if you're crazy, point to any of the two sisters you like [you'll do whatever the hell you want anyway]". And then you marry the sister pointed to (the pointing, of course, actually being done via some chosen mapping of "yes" and "no" to the two sisters). Every solution is just this, worded one way or another.

Last edited by Indistinguishable; 11-21-2009 at 05:19 PM.

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