Brainteasers tend to follow a particular pattern. The amount of information the various Alices and Bobs and Charlies have seems at first glance to obviously be inadequate to the task they're given. Then a closer examination reveals that there's some sublte extra scrap of information, and you can generally assume than that the answer is to use that to trace out the possibilities to find the actual answer.

So knowing that, I just looked for what extra, non-obvious piece of information A and B had. It seems impossible to guess what the results of a coin-flip are in another room. But the players also know the result of their own flip, and so basing a strategy on that extra info is certainly going to be the answer. From there, since there are so few combinations of such a strategy, getting the right answer is pretty trivial.

I like this brain teaser (and I suspect the reason that Felton likes it as well) because it shows this general pattern for brain teasers very clearly. The "extra" piece of information is pretty obvious with a moments thought. And once you have it, the number of possible strategies using it is small enough that you can see the correct one quickly.

A lot of other brain teasers require a lot of pen and paper work even after you've guessed the "trick" in order to work through all the possiblities. So this is sort of a nice, boiled down "essence du brainteaser".