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The Sage's Hat


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this is of similar style to my Three Philosophers problem, except this one is easier:

The three wisest sages in the land were brought before the king to see which of them were worthy to become the king's advisor. After passing many tests of cunning and invention, they were pitted against each other in a final battle of the wits.

Led blind-folded into a small room, the sages were seated around a small wooden table as the king described the test for them.

"Upon each of your heads I have placed a hat. Now you are either wearing a blue hat or a white hat. All I will tell you is this- at least one of you is wearing a blue hat. There may be only one blue hat and two white hats, there may be two blue hats and one white hat, or there may be three blue hats. But you may be certain that there are not three white hats."

"I will shortly remove your blind folds, and the test will begin. The first to correctly announce the colour of his hat shall be my advisor. Be warned however, he who guesses wrongly shall be beheaded. If not one of you answers within the hour, you will be sent home and I will seek elsewhere for wisdom."

With that, the king uncovered the sages' eyes and sat in the corner and waited. One sage looked around and saw that his competitors each were wearing blue hats. From the look in their eyes he could see their thoughts were the same as his, "What is the colour of my hat?"

For what seemed like hours no one spoke. Finally he stood up and said, "The colour of the hat I am wearing is . . ."

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Where I saw this, the solution was long winded and didn't make sense. I'll clear it up:

The question is, is there 1, 2 or 3 blue hats? If there was 1 blue hat in the room, the person wearing it would see the other two white hats, and announce his hat color instantly. So there is 2 or 3 blue hats. From the other sages' silence, the sage knows he can't have a white hat on, because if he did, another sage would see 1 blue and 1 white hat and know his had to blue because there can't be only 1 blue hat or the wearer of it would know instantly. So another sage would know his own is blue because there has to be at least 2 blue. And nobody has spoken, meaning there is 3 blue hats. And you can think of it this way: all three sages have tied so far, so the king would have to put them all in the same position to make it a fair match?

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an even better explanation (less muddled):

Sage 1 is the one that figured out he had a blue hat first. Sage 2 and Sage 3 are the other sages.

Sage 1 looks at the facts:

* If there was only one blue hat, the wearer of it would know instantly he had it when he sees two white hats. Nobody shouted out as soon as he saw, so there is 2 or 3 blue hats. All three sages would be able to quickly deduce there is 2 or 3 blue hats in the room.

* If I (Sage 1) am wearing white, Sage 2 sees one white hat and one blue hat, and knows he himself cannot be wearing white because Sage 3 would see two white hats and know himself (Sage 3) has a blue hat. There has to be at least 2 blue hats, so Sage 2 would know, upon seeing one white and one blue, that he is the second blue hat. Sage 2 would announce that he has a blue hat, and win.

* But Sage 2 is silent. Meaning the only possible thing Sage 2 can be seeing is two blue hats. That's what Sage 3 is seeing. That's what I am seeing. There are 3 blue hats. We all have blue hats, the king's test is whoever can figure it out first.

So Sage 1 stands up and says "I have a blue hat"

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..blue

and here's why

Lets say man A answered the question

lets just say A had a white hat

then B would see one blue hat and one white hat, but B's hat can't be white because then C would have said that his hat was blue, because there has to be at least one blue hat. But since C didn't speak up he doesn't see two white hats, but B sees one, so I must have a blue hat.

But B didn't say anything either.

So that means A had to have a blue hat

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Let the three sages be A, B, & C.

'A' sees the two blue hats on the heads of B & C. He waits and thinks, "What B & C are seeing..... if I have white hat, both will see one blue hat on each others head and one white hat on my head. Then, either will wait for answer from the other, both assuming the possibility of a white hat on their heads. But only I know that both of them have blue hats, so nobody will answer. Therefore, after hearing no answer from one another, B & C - both will be able to know that they did not have white hat, and quicker one will be able to tell his hat's color is blue.' But what if I had a blue hat...? In that case they will be thinking in the similar way as I am thinking, waiting response from either me or the other one. "

After waiting for a calculated time, when 'A' hears no response from B & C, being the quickest, 'A' calls out first, "My hat's color is blue."

Edited by bhramarraj
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