Three Princesses Riddle
Posted by Rob Herman at September 28th, 2006
A riddle for you today, and another one next time. There’s a method to my madness this time, though, because I want to talk about Hint 0 and why it’s so important for the kinds of riddles I like. (Hint 0: A solution exists.) Today’s riddle is much easier, I think, if you take Hint 0 into account. It’s even more important for the follow-up.
As a reward for your valiant service as a knight, the King offers you the hand of one of his daughters in marriage. The oldest daughter always tells the truth. The middle daughter may tell the truth or lie, arbitrarily. The youngest daughter always lies. Of course, you don’t know which daughter is which. You want to make sure you marry either the oldest daughter or the youngest. (If you marry the liar, at least you know where you stand.) You only get to ask one yes-or-no question of one daughter. How can you use this question to pick a daughter who meets your goal?
Clarifications: If it makes your questions more convenient, you may assume the daughters have names, say, Anne, Beth, and Carol. You cannot ask a question that the daughter might not know the answer to. For instance, you cannot ask “What would Beth say if I asked her if she is the oldest daughter?” because if Beth is the middle daughter, the other two wouldn’t know.
Hint 1: Strictly speaking, you don’t need the oldest/middle/youngest designations, but it makes the question you ask a lot more elegant.
Ask Sister A if Sister B is older (or tells the truth more often) than Sister C. If the answer is yes choose C, if the answer is no choose B. After the wedding you will still have to figure out if you have the oldest or youngest sister.
I figured this out on a long car trip back form Stowe with my wife. Sadly, I cannot think of a way to figure out which sister is the middle one. The marriage to her would be the most interesting.
With one yes/no question, you definitely can’t figure out which is the middle one. There are six (three factorial) ways the daughters could be arranged. Each daughter is in each position in two of the arrangements. With one yes/no question, there is no way to eliminate more than half of the possibilities. Even if you could ask the King (always truthful or whatever) one yes/no question, it still can’t be done.
The question is, “What would the youngest daughter say if I asked her who is the middle child?”
The youngest daughter, since she always lies, would either say she is, or the oldest daughter is. But since the question asks her what she would say, and since she always lies, she would say the middle child.
The oldest daughter would answer the question, just as the younger daughter would, answering the middle child.
The middle child would either answer honestly and the same answer the other two did, and name herself. Or, she lie and give an option of the two others.
So, you know, if the daughter’s answer is one person other than herself, then marry her or the one besides the one she names.
If the daughter’s answer is herself, then you know she is the middle child and is answering the question honestly so choose one of the others.
If the daughter’s answer has more than one choice, then you know she is the middle child and is lying, so of the two names she gives, choose one.
This is assuming that when answering the question, the daughters will give all the possible answers AND that the daughters know who is who and which always lies, always tells the truth, etc.
Victor, that isn’t a yes-or-no question, but I think it works otherwise. The assumption that the daughters know who is who is certainly valid. “Giving all the possible answers” is made unnecessary by the requirement of yes/no question. (Beaker’s solution works.)
I agree with Beaker, once you accept that you can not pick the one you are asking, you then only need to force the other two to agree by using a question that takes advantage that the lying sister has to lie.
You now need to decide who you are going to ask the question? May as well choose the one you find least attractive since you can’t pick her :)
Mark: Yes, that’s right; welcome to Rule 0; and that’s a great way I hadn’t thought of to get more utility out of the question! ;-)