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HatsPuzzle - Hats A L TEX le hats Daniel A Graham...

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Hats * L A T E X file: hats — Daniel A. Graham <[email protected]>, June 16, 2005 Imagine a class with n very logical girls sitting in a circle. The teacher enters the room with a bag of hats, informs the students that each hat in the bag is either white or red and then, walking behind the girls, places a hat on each girl’s head. No girl can see the color of her own hat but each can see the color of every other girl’s hat. We suppose, in fact, that all the hats are red. Now the teacher states that she will give a prize to any student who can correctly identify the color of her own hat and a large enough penalty to any girl who specifies the wrong color to discourage guessing. The teacher then progresses repeatedly around the circle, asking each girl in turn whether she would like to state the color of her own hat or pass. Will any girl ever be able to state the color of her own hat? No, as we shall see, each girl will forever pass. Now suppose the teacher announces to the class that at least one of the hats is red and then once again progresses repeatedly around the circle asking each girl in turn whether she would now like to state the color of her own hat. Will any girl ever be able state the color of her own hat? Yes, every girl will, in fact, eventually be able to state the color of her own hat. How the announcement of a fact that was already obvious to every girl produces such a dramatic change in the outcome is the subject to which we now turn. To facilitate a graphical exposition, we consider the case in which there are three girls, let 0 denote white and let 1 denote red and use the binary number c 1 c 2 c 3 to denote the state in which c i is the color of the hat worn by the i th girl. Thus 001, for example, denotes the state in which the first two girls are wearing white hats and the third girl is wearing a red hat. Since we are supposing that all the hats
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