Quote:
Originally Posted by orlok
Well I did say it came to me in a dream...
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It doesn't sound like a good dream
I'm not a mathematician, but I have a long education containing some mathematics, and when I dream about numbers it's usually nightmares...
However, since we're on a subject that in the wide sense concerns mathematics, lets continue with that - my question is not about numbers or sequences, but, if there are any mathematicians out there, this task can be solved by
mathematical induction. For all non-mathematicians out there, this task can be solved with logical reasoning.
This is not a trick. The task has a solution, and it can be found by using logic. I admit that the solution may be a wee bit farfetched, but here goes:
Imagine a room with five people in it, all of them are wearing a red hat. None of the persons know the color of their own hat, but they can see the color of all the other hats. There are no mirrors or reflecting surfaces in the room they can use to look at their hats. They are strictly forbidden to communicate about the subject of hats and color, so asking each other (in
any way) about the color of the hats they are wearing is not an option. The persons can only leave the room if the know
for certain the color of their own hat. A person can only leave the room at a whole hour (i.e. 12.00, 13.00, 14.00 etc.).
Imagine furter: The time is now 11.55. A person comes in the door, says: "I can see a red hat" and leaves the room.
No other person enters the room and the five persons does not receive any more information about their hats. We have to assume that all the persons in the room reasons in the same way, that they observe the same things and draws the same conclusions.
Question:
When can the last of the persons leave the room?
A short summary of the problem:
- None of the five persons know the color of their own hat, but they can see the color of all other hats except their own.
- They know they can only leave the room when they know for certain the color of their own hat. Guessing is not allowed.
- They cannot ask each other in any way or communicate with the outside world to discover the color of their hat. There are no mirrors or other tools available. The only way to find out is by deduction.
- They can only leave at a whole hour (12.00, 13.00, 14.00 etc.)
- At 11.55 there is at least one red hat in the room
- All persons observe the same things, draw the same conclusion and reasons the same way. I.e., if person A thinks of something, he knows that all the other persons knows/thinks exactly the same.
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?