Quote:
Originally Posted by norway1456
Only we, who are to solve the problem, knows that all the five hats in the room are red. The persons in the room must deduce, from the information given above, the color of their own hat. The problem has a logical solution. When can the last of the persons leave the room?
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This is a different statement of the prisoners with coloured dots on their foreheads problem.
The interesting thing is that it seems that none of the prisoners have been given new information, as they all already know that there are red hats in the room.
What is more, since they can each see four red hats, they all know that everyone else in the room can see at least three red hats.
What they do not know is that everyone else can also see four red hats, they gain this knowledge with the passage of time, by reasoning this way:
If I were seeing only blue hats, I would be able to leave at 12:00, as I'd know that my hat was red. But I can see some red hats, so I can't leave at 12:00.
If I were seeing three blue hats and one red hat, and my hat was also blue, then the person with the only red hat would leave at 12:00, as they could not see any other red hats, and I could then leave at 13:00. But I can see more than one red hat, so I can't leave at 13:00.
If I were seeing two blue hats and two red hats, and my hat was also blue, then the two people with red hats would leave at 13:00, as each of them would only be able to see one red hat, and I could leave at 14:00. But I can see more than two red hats, so I can't leave at 14:00.
If I were seeing one blue hat and three red hats, and my hat was also blue, then the three people with red hats would would leave at 14:00, as each of them would see two blue hats and two red hats, and I could leave at 15:00. But I can see more than three red hats, so I can't leave at 15:00.
I am seeing four red hats. If my hat was blue, everyone else would see one blue hat and three red hats, and would all leave at 15:00. But they haven't left. So my hat must be red, and we can all leave at 16:00.