But, when the fielder throws his mitt and touches the ball, the ball is in fair territory and thus the ball becomes fair the moment his glove touches it. Yes, the batter is awarded third base, but if he can make it home on the play he's entitled because the ball is still considered to be in play. I don't think it's true that a ball is not fair or foul until it reaches 1st or 3rd. It's fair until it goes foul before reaching the base.
So, the batter in this riddle did hit a fair ball.
Here are the rules I think would govern:
Definition: A FAIR BALL is a batted ball that settles on fair ground between home and first
base, or between home and third base, or that is on or over fair territory when bounding to
the outfield past first or third base, or that touches first, second or third base, or that first
falls on fair territory on or beyond first base or third base, or that, while on or over fair Rule 2.00
territory touches the person of an umpire or player, or that, while over fair territory, passes
out of the playing field in flight
Rule 7.05 Each runner including the batter-runner may, without liability to be put out,
advance—
Rule 7.05 c Three bases, if a fielder deliberately throws his glove at and touches a fair ball. The
ball is in play and the batter may advance to home base at his peril.
I think this makes sense. It's kind of like a delayed call in hockey, where the ref lets the play happen before stopping play so as not to penalize the batter. Perhaps there is a rule that is an exception to what I've outlined above. Regardless, thanks for posting it. It's an interesting scenario, and I'll try to come up with another one.