More Bs means it's softer, so more graphite will get onto your paper when you draw a line, which makes it darker.
More Hs means it's harder, so less graphite. The advantage is that it doesn't get used up as quickly and you can draw finer lines, although the latter is kind of a given either way, since you're using a mechanical pencil.
The problem is that it sounds like a riddle. In a riddle, you're traditionally supposed to work within the rules that you've been told. So, not thinking outside the box here is not an indication that the person isn't capable of doing so.
Of course, if I encountered this problem in real life, I'd ask Carol from accounting to check the other room, while I flip the switches. But my instinctive answer was that it is not possible, because I assumed it to be a riddle and the provided rules did not allow a solution.