Eight logicians stand one behind the other facing an opaque wall such that 3 are on one side and 5 are on the other. None of the logicians can turn around or see beyond the wall.
Each wears a black or white hat as shown below; '|' represents the wall; capital letters are used to identify the logicians; b and w refer to black and white respectively:
w b w | b w b w b A B C | D E F G H
Each knows the location of the others and the quantity of each colour of hat. They also know that all hats of same colour are not on the same side. Logician should announce the colour of owned hat once sure.
Considering that all logicians think at the same speed, who will be the first to declare having which colour? How many can come to know the colour of owned hat and in what order?
Considering that all logicians think at the same speed, who will be the first to declare having which colour? How many can come to know the colour of owned hat and in what order?
| See The Solution | Submitted by Salil |
| Rating: 2.8000 (5 votes) |
First to declare
|
 | Comment 1 of 8 |
G should be the first to declare that he's wearing a white hat. He sees two black hats in front of him and knows that if he were wearing a black hat, then H would know his own is white (since they all know that all four hats of a given color are not on the same side of the wall). When H doesn't immediately declare himself to be wearing a white hat, then G will know his own is white.
At that point, F will be able to declare that his hat is black. Once again, he knows that since E and G are wearing white hats, if his own hat were white, then H immediately would have declared black.
E comes next, for the same reason: He sees D's black hat and now knows that F has one, too. If his own hate were black, both G and H immediately have declared white. So he knows his hat is white.
Now that D knows there are two white hats behind him, he too knows that if his own hat were white, then H immediately would have declared black. So D knows his hat is black and can speak up accordingly.
At the same time, B sees C's white hat and also now knows that there are at least two white hats on the other side of the wall. If his own hat were white, then A would know that his is black. Since A remains silent, B knows that his hat is black and declares as much.
Meanwhile, C knows that the only way B could know that his hat was black is if his own were white. (If C's hat were black, then A would know that his is white - seeing two black hats in front of him and knowing that the other two are on the other side of the wall.) So C declares his hat to be white.
And I think that's where it stops. A and H both know that one white and one black hat remain, but neither has any way of knowing which he is wearing.
Maybe?
At that point, F will be able to declare that his hat is black. Once again, he knows that since E and G are wearing white hats, if his own hat were white, then H immediately would have declared black.
E comes next, for the same reason: He sees D's black hat and now knows that F has one, too. If his own hate were black, both G and H immediately have declared white. So he knows his hat is white.
Now that D knows there are two white hats behind him, he too knows that if his own hat were white, then H immediately would have declared black. So D knows his hat is black and can speak up accordingly.
At the same time, B sees C's white hat and also now knows that there are at least two white hats on the other side of the wall. If his own hat were white, then A would know that his is black. Since A remains silent, B knows that his hat is black and declares as much.
Meanwhile, C knows that the only way B could know that his hat was black is if his own were white. (If C's hat were black, then A would know that his is white - seeing two black hats in front of him and knowing that the other two are on the other side of the wall.) So C declares his hat to be white.
And I think that's where it stops. A and H both know that one white and one black hat remain, but neither has any way of knowing which he is wearing.
Maybe?
| Posted by Jyqm on 2006-07-02 11:34:55 |
