I couldn't find a similar thread (okay, I didn't look very hard), so I'm starting a new one. This is the place to go if you can't remember the title of a book or a story, but you can remember enough that maybe someone else can track it down for you.
This is what I'm looking for: (I thought it was one of the stories from 1001 Arabian Nights, but, if it is, it's hiding from me well.) It's a story about a king and two young men who are either suitors for his daughter, or otherwise in line for the throne, both vying for his favor. One day, the king is drowning in his pool, and the first suitor happens to be near and saves his life. The second suitor is cagey, though, and he realizes that saving the kings life is a) not something he can easily orchestrate and that b) the king will eventually resent the life-saver for making him feel in debt. So the cagey second suiter fakes his own drowning so that the king can save him, and thus feel powerful. His plan works, and he ends up with the throne.
I would so much appreciate any help with getting a lead to track this thing down!