A logic problem by Daniel Patten
Alan, Barry and Carl are lined up facing a wall in prison, and the guards them that they are going to place a hat on top of each of their heads; either white or back. There are a total of five hats; three black and two white. They are told if one of them can guess the color of his own hat, they will all be set free. Carl is at the back and can see the colors of Alan's and Barry's hats, Barry can see the color of Alan's hat, and Alan can see nothing. The hats are placed on their heads. Carl says 'I don't know the color of my hat'. After a few seconds, Barry says 'I always don't know the color of my own hat'. Alan then says, I know the color of my hat. He gives the correct answer, and the three men are released. What color hat was Alan wearing?
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1 solution
It is perhaps a good idea to view in our own minds eye, what each of the men see, and how the mindset of each of them changes as we move up the queue. As Carl is the man at the back, he can see the color of the two hats in front of him. There are a total of four possible cases that he is seeing: - both Alan and Barry are wearing black hats - both Alan and Barry are wearing white hats - Alan is wearing a white hat, and Barry a black - Alan is wearing a black hat, and Barry a white Since Carl doesn't know the color of his own hat straight away, we can rule out the possibility that both Alan and Barry are both wearing white hats, as this would have been enough to tell Carl that he was wearing a black hat. Knowing this, Barry says to himself that either he and Alan are either both wearing black hats, or one of them is wearing white and the other black. He of course, can see the color of Alan's hat, and he also says that he doesn't know the color of his own. The two possibilities of what Barry can see is obviously that Alan is wearing a white hat or a black hat, but had he seen that Alan had a white hat, he would have then been able to identify his own hat as black. This immediately lets Alan know that the color of his own hat must be black.