The three wise men problem

A

arfa brane

call me arf
Valued Senior Member
You maybe know this one:

A king summons three of the wisest men known in his realm, and tells them he wants to test their wisdom.

He explains they will be led to a room in which they must sit in a circle facing inward.
He then tells the three that each of them will be blindfolded and a hat placed on their head.
Each hat can be only white or red. After removal of the blindfolds, the first of the three to say what color hat he is wearing will be ruled the wisest.

At this point the king usually tells them some additional information, but I'm leaving it out.
The problem changes now to this:

After having the hats placed while blindfolded, each then has their blindfold removed.
One of the three doesn't open his eyes.

What is he thinking?
 
arfa brane said:
What is he thinking?
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That he can deduce the answer based solely on the answers of his compatriots?
No, that doesn't make sense. He could just as easily do that with his eyes open...
Unless he's blind. Then there's no point in him opening his eyes.

There isn't a stipulation that wrong answers result in them being put to death, is there?
 
arfa brane said:
What is he thinking?
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That he's blind, and opening his eyes won't make any difference.
That a truly wise man does not seek to be ruled "wisest".
That he'd rather catch up on some sleep.
Or that the hat they're being asked about is the one the King is wearing, which he probably saw already, so has no need to open his eyes again, although the King may have changed his own hat.

If the test is fair, though, and the request is with regard the colour of the hat that the wise man himself is wearing, then all hats will be of the same colour, and all of the men will be able to see.
If it is stipulated, for example, that there is at least one hat of each colour then the only person who could speak out knowledgably would be the one that saw two hats of the same colour.
 
The wise man with his eyes closed is thinking thusly:

If I keep my eyes closed, it could give me an advantage. The king has not said all three hats will not be the same color, only that they can each be red or white.

If I assume the other two wise men have their eyes open and can see two of the hats, the solution comes down to whether they can know the color of their own hat, so I should wait for them to say so.

The king has summoned us here, and I can assume he doesn't want to play a trick on the three wisest men in his kingdom.

Sarkus said:
If the test is fair, though, and the request is with regard the colour of the hat that the wise man himself is wearing, then all hats will be of the same colour, and all of the men will be able to see.
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Yes, this puzzle turns out to be unfair for two of the wise men.
Kings, huh? What's a wise man to do?
 
Suppose the king decides the fairest test is with all three hats the same color. Then why tell them each hat is red or white?
Suppose he thinks having one hat red, the other two white, is fair. Why not tell them that detail?

So the wise man with his eyes closed, in fact is faced with the test being a test of the wisdom of the king.
 
Let's say the problem was properly set, so some answer would be the right one and wisdom not involved.
He could be thinking that by closing his eyes he screws up the other two - they can't answer for sure without information from him.
 
iceaura said:
He could be thinking that by closing his eyes he screws up the other two - they can't answer for sure without information from him.
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An interesting ploy.

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I note, by the way, that the question at the end is not "How does he solve the King's puzzle?" which is what we would assume is the goal of this puzzle.

"What is he thinking?" does not suggest, in way way, he is thinking about how to solve the puzzle.

For all we know, he's thinking: "Who would win in a fight? Superman or Mighty Mouse?"

I'm not just being flippant. I'm suggesting that it may be a trick question, designed to imprint an assumption on us.

IOW, as the puzzle is worded, there are no wrong answers.
 
DaveC426913 said:
IOW, as the puzzle is worded, there are no wrong answers.
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Technically correct. However, a wise man might gauge that against telling the king that "answer".
iceaura said:
Let's say the problem was properly set, so some answer would be the right one and wisdom not involved.
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No, wisdom would still be involved. One of the wise men, in your "properly set" scenario, has to decide that wisdom is not involved in determining the answer (maybe it's merely a combinatorial problem).
Would that be a wise decision?
He could be thinking that by closing his eyes he screws up the other two - they can't answer for sure without information from him.
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Which information must surely be limited to what he says. He can only really say that he knows the color of the hat he's wearing, or he can say he doesn't know.
 
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arfa brane said:
Technically correct. However, a wise man might gauge that against telling the king his "answer".
No, wisdom would still be involved. One of the wise man, in your "properly set" scenario, has to decide wisdom is not involved in determining the answer.
Would that be a wise decision?
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Granted. I realized that, afterward, stating they are wise men behooves us to assume they are being wise.

So, can we move forward assuming it is not a trick question? That, in the context of the puzzle presented to him, he is ostensibly acting wisely?
 
The thing about this puzzle (it's an old one) is that it has to be unfair so only one of the three contestants can determine the color of their hat, through logical exclusion of the other possibility.

Let's suppose the wiseguy with his eyes closed, is thinking about how the king might have arranged for each of them to be wearing the color hat he decided they would, i.e. how does he allow freedom to exist in an otherwise already determined outcome?

So the wiseguy thinks about say, the king arranging for each hat to be randomly chosen. In that case, there are only two possible outcomes (modulo two colors): 1. all three hats are the same color, 2. one of the hats is a different color than the two remaining.

If 1. is the actual situation, then none of the three wiseguys has a chance--they all got told the hats are red or white, and each can see two hats the same color.
If 2. is the situation, then one of the three wiseguys has an advantage, but only if he can decide the other two can't tell their own hat color, whether they actually say so, or say nothing for long enough (for a wiseguy) that it's obvious.

Hence, the dude with his eyes closed has to reason that the king has not made all three wear the same color hat. Which means, when he opens his eyes, his hat must not be the color of the two he can see.
 
My version of this puzzle relies on at least one of the wise men reasoning that the king telling them the hats are either red or white means the hats aren't all the same color, because otherwise the test isn't fair.

But the king doesn't tell them something that the original puzzle does, which is that at least one of the hats is red. The king could also say at least one hat is white. So the king tells them some specific information about the color of at least one hat.

In that case the wise man who keeps his eyes closed can announce he knows what color his hat is without opening his eyes. He now knows that at least one, at most three hats are red. If neither of the other two wise men can solve the problem then he can be certain his hat is red.
 
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