The Three Princesses (distinguishing truth-teller with 1 question)
Pick a random prince, then ask him:
Is he [pointing to one of his brothers] older than him [pointing to the second one]?
If you receive "Yes" as an answer, choose the second brother, otherwise choose the first. This ensures you will not marry the middle one.
Explanation:
If the prince you asked is the eldest, he will answer truthfully, and you will pick the youngest.
If the prince you asked is the youngest, he will lie, and you will pick the eldest.
If the prince you asked is the middle one, he will answer randomly, and you will pick one of the two others, as desired.
Edit:
More information about the many variations of this puzzle can be found by searching for the term "Knights and Knaves", which is the name logician and author Raymond Smullyan used to describe these problems.
Of particular interest is a problem known as The Hardest Logic Puzzle Ever, which has the same basic setup as the "princes(ses)" puzzle – lier, truther and random answerer – but with more information needing to be extracted and the additional complication that instead of "Yes" and "No", equivalent words in some esoteric language are used and it is unknown which word means which.
Ask the middle one if the youngest is more truthful than the eldest. If the answer is "no" then marry the youngest, otherwise marry the eldest.
That way, if the middle one is the liar, he is guaranteed to marry the most truthful of the remaining two, which is the 100% truthful princess. If the middle one is the truthful one, then he is guaranteed to marry the least truthful of the remaining two, who is the habitual liar. If the middle one sometimes tells the truth and sometimes lies then it doesn't matter which of the other two he marries (for the purposes of this riddle...).
This is just the answer I gave in the comment above, removing the mistake pointed out by Theo Buehler (and the implicit ageism in assuming the prince would rather marry one of the younger two).
Label the brothers arbitrarily as $A$, $B$, and $C$ (as mathematical parents would). We can describe any strategy we might adopt as three pieces of information:
- A question to be addressed to $A$. (Since the labeling were arbitrary, we assume $A$)
- The brother we will marry if we receive "Yes" as an answer
- The brother we will marry if we receive "No" as an answer.
You cannot marry the brother you ask the question to.
To show this, suppose that we marry $A$ if he responds "yes". It is possible that $A$ is the middle brother. Similarly if we marry $A$ for a response of "no" - it is, in fact, impossible for us to determine whether the person we ask the question to is the middle brother, since if it were, we could receive any answer - and in particular, we could receive the same answer as if he were not
Thus, we can assume, without loss of generality, that we will marry brother $B$ if we get a "yes" and brother $C$ if we get a "no" (since we should obviously not have chosen the brother we marry before hearing an answer!). Moreover, this means that we can assume that $A$ is not the middle brother, since if he is, the strategy works regardless of the question.
There are a number of questions that will let us determine, assuming $A$ is the youngest or the oldest, whether $C$ is the middle brother. For instance
Is $C$ older than $B$?
functions, since if we get the youngest and a "yes", we know $C$ is the middle brother and we should marry $B$ - and we can work out the other cases similarly. In particular, any question with the following truth table functions: $$\begin{array}[ccc]. &&\text{$B$ is the middle brother}&&\text{$C$ is the middle brother}\\\text{$A$ is the youngest}&&\text{true} &&\text{false}\\\text{$A$ is the oldest}&&\text{false}&&\text{true}\end{array}$$ To be very blunt we could ask $A$:
Are you the youngest exclusive or is $C$ the middle brother?
Another question, along a different line, would be:
If I asked you if $C$ was the middle brother, what would you say?
Since we essentially force the brother to tell the truth - but this is perhaps further from the spirit of the question.