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Famous Logic Puzzles



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?

26 comments:

Secret Squïrrel said...

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.

Anonymous said...

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

Secret Squïrrel said...

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.

Anonymous said...

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.

-------------------------

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.

Anonymous said...

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.

Anonymous said...

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.

Anonymous said...

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.

Secret Squïrrel said...

@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

Anonymous said...

@SECRET SQUIRREL...ur explanation s very clear and the only soln 2 tis prob. Sad, tat the other 2 r not able 2 understand the logic behind it..good thinking..way 2 go secret squirrel..

tsewang tamchos said...

@anonymous

your explanation
"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?"

this is blunder!!!

proof:-

put A=B (or A=C)in your explanation. and you wud get answers as B and C.
my point here y cannot B reason that other two are not able to ans n hence he is unique..bla bla.


@secret squirrel.
the wordings of the puzzle were
"A king wants an advisor and comes to ask the 3 wisest sages"...'3 wisest sages'implies all were equally skilled.and each of them were looking at other two who were wearing blue hats.so situation is same for all .y only A is able to surmise that he is wearing a blue hat.



well i think the question is wrong/there is no answer.

Anonymous said...

@tsewanu hav concluded wrongly abt the 3 wisest men..for eg: if a teacher calls out for the 3 best students of the call, doent mean tat they all r equal best, BUT they r the top 3 students of the call,one ranked above the other.

sarat said...

@tsewang tamchos
u r explanation "and each of them were looking at other two who were wearing blue hats.so situation is same for all "

is essentially faulty.It is written that only Sage A sees that the other 2 are wearing blue hats.That's the reason why only A is able to surmise that he is wearing a blue hat.

And Secret Squirrel's explanation is spot on. I don't get why you can't understand such a simple and elegant solution.

Ståle said...

Secret Squïrrel, in his/her masterful demonstration, said:

"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."

We also know this is not the case because the original puzzle states that Sage A sees two blue hats, right :)

One aside comment: it seems to me the king himself is somewhat of a sage to have devised this test. I mean, did he decide not to go for only one or two blue hats just on a whim, or did he anticipate the more complex reasoning Sage A ended up performing, allowing him to demonstrate his inductive prowess?

Les Boys said...

Maybe I'm missing something but as far as I can see the King only said that the sages were wearing either blue or white hats and that at least one of them was blue. The king did not say that any of the hats are actually white. Therefore, whichever colour you choose can only be a guess :)

Anonymous said...

@tsewang tamchos
The phrase "three wisest sages" does not imply that that they are all equally skilled any more than the phrase "three tallest people" implies that they are of equal height.

The king is trying to determine which of his 3 wisest sages is actually the wisest (or best logician, really).

Anonymous said...

I just have to say this.. Does the King ever tell Sage A that he is correct? His hat can be either Blue or White, regardless of the 'not speaking for hours'. This is one of many puzzles that have no 100% answer. I've come across many puzzles where there is no 100% answer to the puzzle, like this one.

Okay here's my reasoning..
If Sage A has a white hat. B and C would both see a blue and white hat.
-We know that.
So B and C don't say anything because their hat can be either White or Blue.
-We know that.
Sage A see two blue hats.
-We know that.
So Sage A can come up with the conclusion that he has a White hat.(Simply reasonable because the King says they're wearing either a White or Blue hat.)
-We know that.
But do we know if he's lying?
If the King is lying to them...The sages would KNOW that and speak up, but that may aquire some 'LONG THINKING' to come up with the solution that the King may be lying. So..

If Sage A was wearing a white hat, the Kings statement will be true. If he was wearing a Blue hat, the Kings statement will be false.

So Sage A would assume that the King is telling the truth and say that he has a white hat. BUT..If he knew the King was telling the truth that everyone was wearing either a white or blue hat, a wise sage would believe him and say his hat is white, but why after a long time? In Sage A's mind he is thinking if the King is lying or not, that's why he is taking so long to answer. This is what i think that he thinks...

'If i stand up and say that my hat is blue, he will know that i think he's lying, and probably have my head for it. If i say that I have a white hat, he will know that I think he's not lying.

Hmmm.. If i HAVE a blue hat, Sage B and C would be at the same spot as me.. If i am wearing a white hat, to B and C, the King's statement will be true..-lightbulb- THATS WHY THEY DIDN'T SPEAK! So the statement MUST be true and therefore i am wearing a white hat. HAH! The King will be so proud! Wait... Maybe they might have gotten the same conclusion if they were as smart as me they would have got to this conclusion also... IM WEARING A BLUE HAT!! No...The King is telling the truth no doubt about it! ITS WHITE!! AGH!!! I am going to speak NOW!! HE'S TELLING THE TRUTH, ITS WHITE!'


-That's my thought of this puzzle..


If the King is telling the truth, that hat is white. If the King is lying it would be blue, green, black, BROWN, whatever color you want it and Sage A would end up dead for not believing the King and Sage C or B would be the advisor.. YAY!!



-But to YOU, is the King lying or not? Which is more reasonable? This puzzle has no exact answer for it. Puzzle solved..

-Call me Krysis. (By the way, I am only 13.)

Anonymous said...

If you didn't understand my reasoning im saying.


This puzzle doesn't ahve enough detail in it to no for SURE the true color of his hat. I congratulate the mastermind to this puzzle. It is a great puzzle and had me thinking for quite a while.

-Krysis

Anonymous said...

There is only one logically acceptable answer which is obtained using simple induction. Secret Squirrel is 100% correct and has used very sound reasoning.

Anonymous said...

If I could type as clearly as him I would make better sense, this puzzle cannot be solved. the puzzle is like this..


Jay is taller than Jen, but John is also tall. Who is the tallest? You can't figure out which one is the tallest without more detail..

-Krysis

mark said...

If I were King, I would have Squirrel as my advisor.

Anonymous said...

What if A and B had blue hats, and c had a white hat, how do they know which hat they have, blue or white.

Anonymous said...

ahh...ambot cin yu.

Anonymous said...

This is a famous problem... The answer is that his hat is blue. The problem can be expanded to any number of people... although typically a series of timed decision points allows a more clear explanation

Anonymous said...

Krysis is righ when he point out that the answer is not 100per cent certain. Subject A blue hat is probably as close as you canget to a certain answer but it is not 100 per cent certain. Example, I have played soccer all my life, put me in the penalty spot with an unguarded goal, what are my chances of scoring. I can say that I can make that even with my eyes close but there is always that factor,maybe I trip,who knows but the bottom line is my skills as good as it might be does not guarantee with 0 percent of doubt a result. Same here as much as the othertwo men are consider wise does not guarantee the answer of subject A.
Krisis is spliting hairs but he has a point. Sorry for my english is not my first language.

Anonymous said...

I'll give it my go at explaining. The point, and what makes it so difficult, is that the solution only comes when one of them realises that none of them are able to guess the answer by what the see, which ironically gives the answer, but takes time. That is why the 2 hours is crucial to the puzzle. It's an elimination of impossiblities rather than a discovery of possibilty.

Firstly, obviously there can't be only one blue hat otherwise it would be too easy, evidently the one with the blue hat would see two whites so his must be blue and he would respond as soon as the blindfolds were removed, so there must be 2 or 3 blue hats.

But if there were 2 blue hats the ones wearing blue would also be able to quickly figure out they had a blue hat, by dint of the other blue hat wearer's silence.

The momentary silence would show that the other blue CAN'T be seeing 2 whites, otherwise he'd say something immediately, so there must be a blue, and as the other they can see is white they themselves must be the blue.

But this doesn't happen either because they can all see 2 blues. They can't figure it out logically from what they see. It's impossible in a sense. And it's this impossibility which ironically leads to the solution. All the other ways are impossible so the only possible solution is that they all wearing blue. Just like Sherlock Holmes.

Anonymous said...

Mathematical possibilities = BBB WWW BBW WWB. subtract "at least one will be blue" = BBB BBW WWB. The color combos will easily deduce an answer, as illustrated by all your posts. The king would be forced to choose if he gave anything other the all blue hats. The hard part for them was to realize the absence of the white hat.

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