I see this as a very interesting problem, even though in some sense it is trivial. I went through the same process, being confused (momentarily) by the wording of it, yet for myself, being pretty experienced with the beat frequency effect (from having been into ham radio), the answer came quickly. I had the experience that made me highly aware of the possibility of getting the same beat frequency from both sides, where one stationary frequency is combined with another one which is variable and sweeping across the point where both frequencies are the same.
However, I can certainly sympathise with how the question leads one to making an invalid presumption that is very hard to go back upon and question, when searching for insight into the problem. The broader question I have, is how can one learn, in the most general way, to recognize when one's own unquestioned presumption might be blocking insight into solving a problem. Incidentally, the wrong presumption here, was where the initial attempt to solve the problem, included the statement "if we assume the frequency of B as 216, on waxing[,] the beat frequency decreases..." . Actually that is not correct. Clearly true at first, but it omits to say that when the beat frequency reaches a certain point (zero in this case) it then increases again -- instead of continuing to decrease (into negative frequency land? :-) ! Of course the effect of absolute value brackets in a formula is what one has to be wary of, and yet, the effect of absolute value may so rarely come into play except in certain very special cases that it is easy to overlook.
I guess the answer is, when stuck, to remember to debug for all possible faulty presumptions. The major effort may be to first carefully identify all the presumptions. (Good luck on that..) Then one can try to think of all possible, if seemingly unlikely, variations of them, and work out the various results. Turn over enough rocks and likely there will be a bug under one of them.
And of course, one can specifically learn the rule of watching out for the implications of absolute value brackets... :-)
