Okay, I admit this is an odd example of problem solving.
Who really needs to know how to make a half court basketball shot while facing away from the basket and not ever looking at it? Not many people, so what value is there in knowing how? Not much, but there is value in knowing how to figure it out.
Similar problem solving skills can be applied to other situations.
The solution is found partly in having the strength and shooting technique to get the ball into the basket.
The rest of the solution is found in being able to figure out exactly where the basket is, without looking.
This is where the problem solving comes into play.
Tossing the ball over our shoulders with both hands is probably the right approach for distance and loft.
We'd like to have a generous arc so we have a chance of a "swish" as well as a shot off the backboard.
Now, how do we figure out how to properly aim the shot without looking? This may not appear obvious to many of us, but if you're standing at half court, the basket behind you is exactly the same distance as the basket in front of you at the other end of the court.
It's also exactly the same height.
What you see in front of you looks exactly like what you'd see if you had eyes in the back of your head.
If you turn around, you'll see exactly the same "picture" no matter which basket you face, as long as you are dead center on the court and squarely face the basket in front of you.
So, you don't need to imagine the trajectory needed to lob the ball into the basket behind you, you just need to look at it there in front of you.
You can use this same mental picture to estimate how much effort would be required to lob the ball over that distance.
You simply "translate" what you see in front of you to what needs to be done behind you - all without looking at where you're shooting.
That's how I make a half court basketball shot, backwards, without looking.