You are trying to escape a cave, and come to a junction, and must decide to go Left or Right.

There is a man at the junction, and you have time to ask him one Yes/No question.

You know that the man is one of two brothers, one who always answers honestly, and one who always answers dishonestly.

What question do you pose to the man, and which way should you go (based on his answer)?

A correct question (and the man's subsequent response) will not allow you to determine which brother (Honest vs. Dishonest) you are speaking with. In particular, you must find a question (related to which way to go) for which both brothers will answer the same way.

Succinct Answer: Ask the brother the question:

Would your brother answer 'Yes' if I asked him if the exit is to the Left?

Then, if he answers "Yes", go Right; and if he answers "No", go Left.

Explanation: There are two "bits of uncertainty" in this problem: First, you don't know which brother you are speaking to; and Second, you don't know which way leads to the exit. Having only one Yes/No question (and its response), it is information theoretically impossible to learn both of your uncertainties. Luckily, the riddle only requires you to resolve the latter uncertainty (which way to go). However, recognizing this fact at the start is useful, in that you know you need to identify a question that will resolve which way to go, but will not resolve which brother you are talking to. In particular, you must find a question (that pertains to which way to go) for which both brothers will answer the same way. Indeed, this is the key insight: once you've realized this, it is an easier task to come up with a question that all satisfy this property. While there are many such possible questions, each one entails having the brothers do a hypothetical exercise where they pretend they are answering for the other, which essentially blends the answer so that it combines one honest answer and one dishonest answer (which always combine to form an overall dishonest answer, regardless of if the lie comes first or second).

Let's see how this plays out for the specific question posed above: "Would your brother answer 'Yes' if I asked him if the exit is to the Left?"

Case 1: You're speaking with the Honest brother: We further partition the universe of possibilities based on which way the exit is:

Case 1A: The exit is Left: Since the exit is Left, when posed the question "is the exit to the Left?", an honest person would answer 'Yes' while a dishonest person would answer 'No'. Therefore, the Honest brother (to whom you are speaking) recognizes that his brother would answer 'No' if asked if the exit was to the Left; and thus, the Honest brother will answer 'No' to your question (since he knows his dishonest brother would answer 'No').

Case 1B: The exit is Right: Since the exit is not Left, when posed the question "is the exit to the Left?", an honest person would answer 'No' while a dishonest person would answer 'Yes'. Therefore, the Honest brother (to whom you are speaking) recognizes that his brother would answer 'Yes' if asked if the exit was to the Left; and thus, the Honest brother will answer 'Yes' to your question (since he knows his dishonest brother would answer 'Yes').

Case 2: You're speaking with the Dishonest brother: We further partition the universe of possibilities based on which way the exit is:

Case 2A: The exit is Left: Since the exit is Left, when posed the question "is the exit to the Left?", an honest person would answer 'Yes' while a dishonest person would answer 'No'. Therefore, the Dishonest brother (to whom you are speaking) recognizes that his brother would answer 'Yes' if asked if the exit was to the Left; but being Dishonest, the brother you are speaking to will lie about this, and instead insist that his brother would actually answer 'No'. Namely, the honest answer to your question: "Would your brother answer 'Yes' if I asked him if the exit is to the Left?" is 'Yes'; but since he is dishonest, the brother you are speaking to will instead answer 'No'.

Case 2B: The exit is Right: Since the exit is not Left, when posed the question "is the exit to the Left?", an honest person would answer 'No' while a dishonest person would answer 'Yes'. Therefore, the Dishonest brother (to whom you are speaking) recognizes that his brother would answer 'No' if asked if the exit was to the Left; but being Dishonest, the brother you are speaking to will lie about this, and instead insist that his brother would actually answer 'Yes'. Namely, the honest answer to your question: "Would your brother answer 'Yes' if I asked him if the exit is to the Left?" is 'No'; but since he is dishonest, the brother you are speaking to will instead answer 'Yes'.

Inspecting the above four (exhaustive) cases, you see that no matter whom you are speaking to, you should go Left if they answer 'No', and you should go Right if they answer 'Yes'.