# Which Color Hat?

Once upon a time there lived a king who wished to find the wisest man in the realm to be his assistant. He summons the 3 known wisest men to his court and he administers the following test.

He sits them in a circle, facing each other and he says “I’m going to put either a red hat or a white hat on each of your heads.” He proceeds to place a red hat on each of their heads. Obviously they can see each other… but there are no mirrors in the room so they can’t see what’s on their heads. He says “If you can see a red hat, raise your hand.” They all raise their hands. Then he says “If you can tell what color hat you have on, stand up.”

Time goes on, one guy looks at another guy, he looks at the other guy. The other guy looks at him. Finally one guy stands up. The question is how did he know he was wearing a red hat?

### 5 guesses to Which Color Hat?

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• john

because when the king said to raise their hands if they see a red hat they all raised their hand which means they all had
a red hat

• Rool

Everybody raising their hands on the first question means that there are at least 2 RED hats amongst them.

Case 1: 2 RED Hats – 1 WHITE hat
The smartest one among the two reds, seeing 1 red and 1 white in front of him, would’ve stood up the fastest and said that he’s sure he’s red!
Since this was not the case, well, it’s not the case… (case 1 I mean)

That leaves us with

Case 2: 3 RED hats
After some time with no one standing up, the smartest of all 3 would stand up first, knowing that they all had red hats on. (Otherwise, why is no one standing already case 1?)

So in short, I would stand up knowing I have a red hat. And I would be designated as the wisest WOMAN in the realm. And I would overthrow the king at some point and start my own puzzles and expermiment with people!

• Dude

John, you’re answer was very close, but they could have raised their hand if they saw one white and one red hat.

Rool, your explaination is what we were looking for:

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.
So it all comes down to this:
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.”

Rool, you are the wisest woman in the land and therefore, you’re the winner of the day.
Congrats!

• Howard Lyon

Obviously each of the three saw at least one red hat. If any of the three could see one red hat AND one white hat and the person wearing the red hat had raised his hand that person would know the person wearing the red hat was looking at a red hat … his. He could with certainty claim the prize. However, no-one immediately did claim the prize. Which means after an appropriate wait any of the three can conclude that all three are wearing red hats. The first person to figure that out is obviously the smartest of the wizards.

• Rool had the best explanation, super smart. I have a another color riddle about a glass door, here it is: Green Grass Door Riddle