It does work but, I have not the layman’s words or ability to explain it simply. Boolos had to be real bored to come up with this one. It is done in different ways by other bored logicians…It is like a finger puzzle for the mind. 
“If I asked you: “Does ‘da’ mean yes if and only if you are True if and only if B is Random?” in your current mental state, would you say ‘ja’?”
Ask this of A. If A is True, and B is Random, and ‘da’ means yes, A will say ‘ja’.
If A answers ‘da’, at least one of the three premises is false.
If A answers ‘ja’, all premises are true and you have solved the puzzle.
At least methinks this, or something of this ilk, should be your first question.
No, because no matter how many premises are false, both False and Random will answer ‘da’ (and if A is either False or Random, at least one of the premises will be false). Note: I take Random here to be random at speaking the truth or telling a lie, not at saying either ‘da’ or ‘ja’.