Say I pull one ball from each box, and I get two white balls and a black ball. Obviously the box that the black ball came from the box that contains all black balls.
But how can I tell which box contains all white balls, and which contains a mix of black and white balls, without pulling out a second ball?
If the labels are guaranteed to be wrong, then you simply pluck a ball from the bin labeled "mixed". You know it's not mixed, since the label is wrong. Say you draw a white ball. You now know that bin is all white balls. You now have two bins left, labeled "black" and "white". The bin labeled "black" cannot be black, so we know that it is mixed; this leaves the bin labeled "white" as mixed.
These puzzles are more about recognizing fuzzy areas in the problem definition than about any sort of logic. In conversational English, "the labels are all wrong" does not guarantee that each label is guaranteed wrong, only that at least one of the labels is done incorrectly.
Another example: You have a room with one light bulb. Outside the room are three switches. How do you determine which switch controls the light bulb by only opening the door once.
This problem requires you to suspend certain assumptions about the real world (you can't see light coming from underneath the door), while preserving other ones (you can hold the light bulb, and it heats up, it stays hot). Solving the problem is more about finding the right level of fuzzying the problem definition than any sort of test of logic.
Say I pull one ball from each box, and I get two white balls and a black ball. Obviously the box that the black ball came from the box that contains all black balls.
But how can I tell which box contains all white balls, and which contains a mix of black and white balls, without pulling out a second ball?