A king wants an advisor and comes to ask the 3 wisest sages.
He blindfolds them and put the hats on their head. Afterwards,
the king takes off their blindfolds. He tells them that
their hat is either blue or white. He tells them that whoever
can deduce the color of their hat will be his next advisor.
Also he tells them that at least one of the sages will be
wearing a blue hat. The sages can all see each other's hats
except of course, their own. Sage A sees that the other 2 are
wearing blue hats.
For hours no one spoke, then Sage A stands up and tells the
king the colour of his hat. What color is it and how does he know?
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8 comments:
A's hat must be blue.
If it was white then either of B or C (who we are told are wise sages and presumably also reasonable logicians) would be able to work out that they have a blue hat.
Here's how: If A's hat is white then B can reason that if his own hat was also white, then C would see two white hats. Since they have all been told that at least one of them will be given a blue hat, C would know immediately that his must be the blue one. Since C has said nothing "for hours" B should be able to conclude quite quickly that his own hat was also blue.
Since B has also not said anything for hours, A must eventually conclude that the premise that his own hat is white is false and that he also has a blue one (although if these sages were any good they would reach this conclusion after a few minutes).
The logic fails if the reason for the other sage's silence is because they have fallen asleep with their eyes open.
sorry^ guy up there but i think his hat is white because he said at LEAST one hat is blue and if both of the other hats are blue then the rest of them would be looking at one white one blue and that would be a 50%/50%shot if they guessed wich wouled explain the silence for hours hi!!!!!!!!!!!!!!!!!peace love happieness
As I said before, if A's hat is white, then B can quite quickly work out what colour his hat is. It's not simply a 50:50 guess just because there are two possibilities. That's like that supposed science teacher from Texas who says that there is a 50% chance that the Earth will be destroyed by a black hole when the LHC is turned on because "either it will or it won't". He completely ignores all of the evidence that tells us that the chance of it happening are vanishingly small.
Likewise A has more information than just the colour of the hat's that he can see. He knows that they are all clear and logical thinkers. If his hat were white B would see one of each colour. B doesn't have to guess because if his (B's) hat were also white, C would know immediately that his was blue. Since C has said nothing, B would then know that his must be blue also (C could follow the same reasoning). So, if A's hat were white, B or C should work out quite quickly what there hat colour is. Since neither says anything "for hours" A can surmise that his hat is also blue.
Sage A's hat is white.
It cannot be blue.
1. The puzzle says that "Sage A sees that the other 2 sages are wearing blue hats." So both sage B and sage C wear blue hats. That's a given.
2. Neither B or C can deduce the color of their own hats, and therefore, they remain silent.
Here's why:
If A's hat were white, B would see C's blue hat, and see A's white hat. B's hat could be either blue or white.
If A's hat were blue, B would see C's blue hat, and A's blue hat. B's hat could still be either blue or white.
If A's hat were white, C now would see B's blue hat, and see A's white hat. C's hat could be either blue or white.
If A's hat were blue, C would see B's blue hat, and see A's blue hat. C's hat could still be either blue or white.
So Sage B and C cannot speak up for sure.
3. After a while, when Sage A sees that neither sage B or sage C speak up, (and they speak up if they could solve the ridde), sage A reasons that neither sage B nor sage C, based on what they see, can deduce the color of their own hats.
Sage A reasons further that since one of the three sages should be able to speak to the king, sage A alone must be the one able to figure it out, since the other two cannot.
There must be something that makes him unique, something that he sees that the other two don't see, something that he sees that makes the riddle solveable for him, while the other two cannot solve it. What does he see?
He sees two blue hats (B's blue hat and C's blue hat). So Sage A again reasons correctly that only he sees two blue hats. Since sage B's hat is blue and sage C's hat is blue, Sage A correctly reasons that his own hat cannnot be blue. (Or else all three sages would see the other two sages wear blue hats).
So A's hat must be white, and so A speaks up.
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Note:
If A's hat were blue, all three sages would be wearing blue, each sage would see the other two wearing blue hats, and there would be nothing to differentiate them, with all three silent. Nobody would speak up and they would all go home.
One more thing in addition to my solution above: the fact the B is silent cannot help C figure it out, nor can C figure it out just because B is silent. Neither B, nor C are in a unique position, as demonstrated by the fact that they each see a blue hat and a white hat. (A's white hat).
So neither B, nor C speak up, for neither can be CERTAIN of their own hat color, and they wisely remain silent.
Work it out , A's hat can only be white.
The answer is blue.
The king hasn't decided which he wants to make his advisor so all the sages have to see the same thing to make it a fair test.
If his hat were blue both B and C would see two people with blue hats and cannot say which colour their hat would be due to the fact that the king did not specify about how many hats would be of each colour.
If his hat were white then, again, neither B nor C can deduce what colour their own hat is as the king didn't specify as I said above.
If sage A had a white hat, then using your words, Sage A would be "Unique" but since the King cannot decide which Sage to chose it would have to be a fair test.
I may be wrong, but after thinking about it, this is the only logical explanation I could reach.
A's hat must be White.
The King tells them that they are wearing a Blue/White hat and that at least one of them are wearing a blue hat.
Since A can see that the other two sages are wearing Blue hats, I think it is quite reasonable to say the sage A is wearing a White hat.
@Anon 2
You make a false assumption - you assume that A must have a unique perspective in order to be the one who solves the riddle. The thing about A that differentiates him from the other two is that he is smarter than they are. This is precisely why the king is conducting the test - he wishes to determine who to appoint as his advisor.
You add in your note that if all three were wearing blue hats then there would be nothing to differentiate them so they would remain silent forever. This is an example of first-level logic.
However, these are the king's wisest men, so we understand that they are capable of thinking about the way other people think, and able to incorporate the other sage's reasoned conclusions into their own thinking, and then repeating this cycle until a solution or impasse is reached.
Look at it this way, if there was only one blue hat then the person wearing it would see two white hats and would know immediately that his was blue. We are told that no-one speaks for hours so we know that this is not the case.
If there were two blue hats, each blue hat-wearing person would see the other plus one white hat. Each blue hat person would wonder why the other doesn't speak up (which they would do if they could see two white hats). The only conclusion to be drawn from this is that it is because the other blue hat person can see one hat of each colour. Therefore, it very quickly becomes apparent to each blue hat person, that their own hat must also be blue. Again, we know this cannot be the case since no-one speaks for hours.
The only remaining possibility is that they are all wearing blue hats which, as someone else has said, is also the only way that this could have been a fair test.
You add in your subsequent post that "the fact the B is silent cannot help C figure it out, nor can C figure it out just because B is silent." However, this is an exercise in inductive reasoning, and incorporating the conclusions of others based on what they see is precisely what the hat-wearers must do in order to solve the riddle. Doing this permits them to gain information about the colour of their own hat.
If my explanations have not been clear enough, you might like to have a look at
http://en.wikipedia.org/wiki/Induction_puzzles
and
http://74.125.153.132/search?q=cache:gyFWyfiFjNgJ:www.uamont.edu/FacultyWeb/Wegley/Logic%2520Hats.ppt+blue+hat+white+hat+puzzle&cd=7&hl=en&ct=clnk&gl=au
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