The prof. misremembered the tricky variant of the problem and posed instead a very easy one instead.

The trickier problem is to discern in a single question some arbitrary bit of truth about the world, for example, which of 2 paths leads to the truth teller village (given that exactly 1 does).

No matter the topic (of the bit of info to be discovered), the question necessarily must be answered the same way by the truth and opposite-of-truth teller. This implies that the question has a different meaning depending on whether it's received by a truth or antitruth teller. It does so by making some reference to 'you' (the person asked).*

There's a mechanical solution, no matter what you want to discover. You can build a cumbersome question that spells out an arbitrary truth table explicitly.

For example, "Is it true that ((you're a truth teller and X) or (you're not a truth teller and not X))?" Where you plug in for X: "this path leads to the truth teller village". Turning the expression into unambiguous spoken language is left as homework, but can also be done mechanically.

But this is inelegant and doesn't seem clever. You can pose an equivalent (same output under the stated rules), shorter question, like "If I asked you X, would you answer yes?" The answer will be yes iff X. (because if you ask the liar X directly, he'll say no iff X, but he'll lie about that fact).

You can disguise the technique further by relying on other facts that have a different meaning based on the listener, e.g. referring to "your village" also.

Obviously it's impossible to get 2 bits of information out of one yes/no question - for instance, some arbitrary proposition X, and the polarity of the liar/truth-teller.

* The professor's mistake was to make the goal the very simple "find out if they're a truth-teller or an opposite-of-truth-teller". Most people would arrive first at the simple solution "Do you exist?" (or, equally as informative "are you a tree-frog?" or "is 1+1=2?"). It's easy because all we have to do is ask a question about the world with a known answer, and this lets us know the polarity of the answerer. When your goal is to find out something other than their polarity, the question must be crafted to have the same answer no matter their polarity (be "yes" iff X). When the goal is to know their polarity, the question must have a known answer ("yes" or "no") iff they're a truth teller.