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 answers questions with either yes or no, however she pleases, ignoring the question asked.
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 a single yes or no question which you may only address to one of the sisters. After your question, you must select a princess to marry. What yes or no question can you ask which will ensure you do not marry the middle sister?
Its mostly a standard "knights and knaves" scenario, and I typically dislike these sorts of puzzles because the answer is invariably some bizzare meta-question (that is, a question of the form "If I were to ask that person '(insert question here)', what would their answer be?"). However, the 'intended solution' to this problem has no meta-questions involved, actually it is quite elegant. Given the nature of the question, it probably has a bunch of other solutions too, so it could also be interesting to see if anybody posts an answer I haven't seen.
3 comments:
So, thoughts:
Are princesses constrained to yes/no or may they respond "I don't know" because if they can respond I don't know you may not be able to ask them about the truthfulness or ... not-truthfulness of the other two.
Also, what construction of 'randomness' do we have for the middle sister? Does she evaluate the correct answer to the question and then flip a coin in her head and based on that coin flip answer either the accurate response or inaccurate? Or does every time she's asked a question just flip a coin that says yes on one side and no on the other.
That question asks a more interesting question: Does it matter? CAN it matter? And does that answer change if she can say I don't know.
I tried to be at clear as possible about the fundamental ambiguity in the random sister by saying "answers questions however she likes", but I suppose "however she likes" could be either "true or false" or "yes or no". Anyway, the latter is intended, but does not have an effect on the intended solution (though, it certainly can have an effect on other possible solutions). I will adjust the statement of the problem.
As far as "i don't know", let us assume that you are restricted to asking a question that it is certain that the princess is capable of answering. If you allow such an answer, you essentially have the power of a trinary question instead of a binary one, and that is really useful.
So excluding questions which are guaranteed to be answerable by anyone (ie. you cannot ask questions which they may not be able to accurately give yes/no answers) I can only think of one solution. It is pretty nice though.
As for solutions which allow the trinary question issue: This problem becomes exceeding easy, I think.
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