Alice, Bob, Cathy, and David together have three white hats and four black hats. Without looking, they each put on one hat and throw the other three away. They then each look at their friends' hats (no one can see their own hat or the hats that were thrown away), with the following exceptions:
- Alice cannot see Bob's hat.
- Cathy cannot see David's hat.
- David cannot see Alice's hat.
However, each friend believes the others can see three hats.
Alice says, "I know Bob doesn't know his hat color."
Cathy says, "I know David didn't know his hat color before Alice's statement."
David says, "Before Alice and Cathy's statements, I did not know that Alice did not know her hat color."
Given that the four friends are highly logical and made only true statements, exactly which people are wearing black hats?