Three logicians, Albert, Bill and Carl, are wearing hats, which they know are either black or white but not all white. Albert and Bill can each see the other two logicians, but not Carl, who is facing away and cannot see them.

Each is asked in turn if they know the color of their own hat. The answers are:

Albert: "No."

Bill: "No."

Carl: "Yes."

What color is Carl's hat and how does he know?

(Taken from http://einstein.et.tudelft.nl/~arlet/puzzles/logic.html)

Because Carl knows his hat color, Carl's hat is black.

There are three possible arrangements where Carl has a white hat.
(1) Albert's hat is white and Bill's hat is black.
(2) Albert's hat is black and Bill's hat is white.
(3) Albert's hat is black and Bill's hat is black.

Under arrangement (1), Bill would have been able to deduce that his hat is black as he sees two white hats and he knows that three white hats are not a possible.  As Bill did not deduce this, this arrangement is eliminated.

Under arrangement (2), Albert would have been able to deduce that his hat is black as he sees two white hats and he knows that three white hates are not possible.  As Albert did not deduce this, this arrangement is eliminated.

Under arrangement (3), both Albert and Bill see one white and one black hat.  Carl, can not see any hat so, as with the other two logicians, is unable to deduce the color of his own hat.  As it is given that Carl did deduced his hat color, this arrangement is eliminated.

As all three arrangements where Carl's has is white is eliminated, Carl's hat must be black.

Posted by Dej Mar on 2008-01-11 10:45:11