Five people are stranded on an island.
A wizard comes and places a magic hat on some of their heads.
At least one of them now has such a hat.
Each hat can be seen by everyone but the wearer.
To remove the hat, those (and only those who have a hat) must dunk themselves underwater at exactly midnight.
The people cannot communicate until after the hats are removed.
They all want to remove the magic hats as soon as possible.
How long does it take the people to remove the hats?
It will take N days to remove N hats.
On the first day, anyone seeing a hat will not know if they have one too.
Anyone seeing no hat on any head will know it must be on his head and remove on the first night.
On the second day, only the two with hats will see one hat on the rest, and they will know to remove them at midnight.
On the N’th day, N people will see N-1 hats and know they have a hat.