One Question for Liar and Truth-Teller

Key idea. Make the answer pass through both dwarfs, so that it contains exactly one lie whichever dwarf you ask.

Such a question exists. Point at either dwarf and ask:

Let $L$ be the statement “the left path leads to the sage”. We check both cases.

• You asked the truth-teller. The other dwarf is the liar. Asked about $L$, the liar would say “yes” exactly when $L$ is false. The truth-teller reports this faithfully, so he answers “yes” exactly when $L$ is false.
• You asked the liar. The other dwarf is the truth-teller. Asked about $L$, the truth-teller would say “yes” exactly when $L$ is true. The liar misreports this, so he answers “yes” exactly when $L$ is false.

In both cases the answer is “yes” if and only if $L$ is false. The answer is therefore always the false answer to “does the left path lead to the sage?”, so you take the opposite of what it says. You do not need to know which dwarf you asked.

Alternative. Ask either dwarf: “If I asked you whether the left path leads to the sage, would you say yes?” The truth-teller answers truthfully about $L$. The liar would lie about $L$, and then lies again about what he would say. The two lies cancel. So here “yes” means left.