The Mad Hatter

Jan 26, 2019

RAY: Once upon a time there lived a king who wished to find the wisest man in the realm to be his assistant. He summoned the three known wisest men to his court, and he administered the following test.

He says to them, “Sit down, you morons. I'm going to put either a red hat or a white hat on each of your heads.” Facing them in a circle and standing behind them, he proceeds to place one red hat on each of their heads. There are no mirrors in the room. Each contestant can see just the other two contestants. The king says, “If you can see a red hat, raise your hand.”

They all raise their hands because they're each wearing a red hat. Then he says, “If you can tell what color hat you have on, stand up.” Time goes by, nothing happens. One guy looks at another guy, who looks at the other guy.

The other guy looks at him. Finally one guy stands up and he says, “I'm wearing a red hat.”

The question is, how did he know that he was wearing a red hat?



RAY: The clue is that for a moment or two, nobody moved. Nobody knew for certain what color his hat was, and that's what told the wisest guy that all of the hats were red. Here's how he figured it out. Each of them had acknowledged seeing at least one red hat. Now, Wiseguy #1 knows he can see two red hats, right? Now, *if* Wiseguy #1 had a white hat on, Wiseguys #2 and #3 would have known what color their hats were.

    Step 1: Wiseguy #1 knows he can see two red hats.

    Step 2: Wiseguy #1 thinks, "Hey, if I were wearing a white hat, Wiseguy #2 would see one red hat and one white."

    Step 3: Wiseguy #1 then thinks, "If I were wearing a white hat, and Wiseguy #2 saw one red hat and one white (and if he were wearing a white hat himself), then Wiseguy #3 would have seen two white hats. So, Wiseguy #3 wouldn't have raised his hand to the first question.

Finally, it hits him. Wiseguy #1 thinks, "If that were true, Wiseguy #2 would be sure that he had a red hat. But since Wiseguy #2 was actually unsure about his hat color, it can only mean one thing: my hat is red."


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