People`s hat puzzle on Island..(awesome logic)

Puzzle:

A bunch of men are on an island. A genie comes down and gathers everyone
together and places a magical hat on some people’s heads (e.g., at least one).
The hat is magical: it can be seen by other people, but not by the wearer of the
hat himself. To remove the hat, you must dunk yourself underwater at exactly
midnight. If there are n people and c hats, how long does it take the men remove
the crowns? The men cannot tell each other (in any way) that they have
a hat.


Solution:
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