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# Masters of Logic Puzzles II. (hats)

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Masters of Logic Puzzles II (hats) - Back to the Logic Puzzles

After losing the “Spot on the Forehead” contest, the two defeated Puzzle Masters complained that the winner had made a slight pause before raising his hand, thus derailing their deductive reasoning train of thought. And so the Grand Master vowed to set up a truly fair test to reveal the best logician amongst them. He showed the three men 5 hats – two white and three black. Then he turned off the lights in the room and put a hat on each Puzzle Master’s head. After that the old sage hid the remaining two hats, but before he could turn the lights on, one of the Masters, as chance would have it, the winner of the previous contest, announced the color of his hat. And he was right once again.

What color was his hat? What could have been his reasoning?

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Masters of Logic Puzzles II. (hats) - solution

The important thing in this riddle is that all masters had equal chances to win. If one of them had been given a black hat and the other white hats, the one with black hat would immediately have known his color (unlike the others). So 1 black and 2 white hats is not a fair distribution.

If there had been one white and two black hats distributed, then the two with black hats would have had advantage. They would have been able to see one black and one white hat and supposing they had been given white hat, then the one with black hat must at once react as in the previous situation. However, if he had remained silent, then the guys with black hats would have known that they wear black hats, whereas the one with white hat would have been forced to eternal thinking with no clear answer. So neither this is a fair situation.

That’s why the only way of giving each master an equal chance is to distribute hats of one color – so 3 black hats.

I hope this is clear enough.

The two losing masters wanted a riposte (Edit: against the winning master), so the grand master showed them 5 hats, two white and three black. Then he said: "I will turn off the light and put a hat on each of your heads and hide the other hats. When I turn on the light you will have equal chances to win. Each of you will see the hats of the two others, however not his own. The first one saying the colour of his hat will win." Then before he could turn off the light, one of the masters (the same one again) guessed, what the colour of his hat will be.

What hat was it and how did he know?

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• 2 months later...

I’d submit the correct color is white since it was the only color to choose from. Scientifically, white is the reflection of all colors in the spectrum while black is the absence of color.

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I’d submit the correct color is white since it was the only color to choose from. Scientifically, white is the reflection of all colors in the spectrum while black is the absence of color.

that's an incorrect statement. if you said light instead of color, that would be right. black is the absence of light, while white is the aggregation of all light in the spectrum. however, when talking about colors, white and black are both valid colors.

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I would suggest you rewrite the question because you state two masters want a rematch and that the grandmaster says I will place a hat on each of your heads (i.e. two) so because of this, white would be the right (equal) answer for two contestants. It is unclear in the setup that the grandmaster would also be wearing a hat.

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its imposible to accaully tell which hat would be placed on your head before it happened... the grandmaster could have made a mistake and placed 2 white hats and 1 black making it obviously and he just happened to guess the right answer and othe combinations... but the best adn quickest way would be to say

you are gunna place the white hat on me... then immediatly after go you are gunna place the black hat on me... making one of his statments right

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i had a slighly different approach using probabilities. Scenario 1: 2white,1black is agreed to be not equal for the 2 white. Scenario 2: 1white,2black; the person(pers1) who saw the other 2 black hats has a 66% prob. of knowing he has white. each of the persons(pers2&pers3) who saw 1white/1black also have a 66% prob. but of knowing they have black. The inequality here is that these people double there probability because they might be pers2 or pers3 each with a 66% prob. of knowing they have black. Therefore the only possible scenario would be Scenario 3: 3black hats! all 3 have 66% prob of knowing they have white. Would you agree???

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• 4 weeks later...
I would suggest you rewrite the question because you state two masters want a rematch and that the grandmaster says I will place a hat on each of your heads (i.e. two) so because of this, white would be the right (equal) answer for two contestants. It is unclear in the setup that the grandmaster would also be wearing a hat.

I agree the puzzle could be written better, but you still misinterpreted it. The "two losing masters wanted a riposte" against the winning master from a previous puzzle. The Grand Master has presumably already proven his superiority. I think it would be clearer to add the "against the winning master" to the puzzle as I did above. Then it should be more clear that there are three participants.

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• 3 weeks later...

black. it all has to do with probability

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• 3 weeks later...

Cool!!

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• 2 weeks later...

black.

losing gms: gm1, gm2

winning gm: gm3

case1: gm1 & gm2 wears white then left are blacks.

case2: gm1/2 is black, gm2/1 is white, gm3 is black-> left is 1black & 1white

--> from gm1 and gm2's point of view, they cannot tell. but from gm3,

since gm1 and gm2 cannot give the right answer, then he could tell that

he's wearing black. if he's wearing white, either of gm1/2 could have had

given the right answer... since gm1/2 did not answer, gm3 will know what

he's wearing.

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• 4 weeks later...

Firstly mavedrive, that is the most unclear garbled mess I have ever read, and I was about to spend no time trying to fathom it out. Secondly, it never gets as far as the grandmaster putting hats on their heads, the winner guesses before the grandmaster gives them a hat.

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• 4 weeks later...
I’d submit the correct color is white since it was the only color to choose from. Scientifically, white is the reflection of all colors in the spectrum while black is the absence of color.

Actually for it to be fair the answer would be black since there are 3 black hats and only two white. If the GM felt like it he could be dishonest and put the oppisite color that the master said on his head so the others would not fight or he could disqulify the smart master. But that wouldn't make the Grand M very grand would it?

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• 2 weeks later...

I’d submit the correct color is white since it was the only color to choose from. Scientifically, white is the reflection of all colors in the spectrum while black is the absence of color.

that's an incorrect statement. if you said light instead of color, that would be right. black is the absence of light, while white is the aggregation of all light in the spectrum. however, when talking about colors, white and black are both valid colors.

No, actually, he's correct.

Those statements really have nothing to do with light. Black is NOT the absence of all light, otherwise, when I'm wearing a BLACK shirt, you wouldn't be able to see it or it would cease to be black when a light shined on it. Black is the absence of all colors.

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Grand master said "I will turn off the light and put a hat on each of your heads and hide the other hats. When I turn on the light you will have (EQUAL) chances to win."

Since we have three contestants and only two white hats then white can not be an option.

Therefore the grandmaster would be placing black hats on all of their heads to be in keeping with his statement.

were these "Ten Gallon" hats? and if so will they be able to use the three remaining hats to measure 29 gallons of water without shorting out the three lightbulbs in the other room?

just a thought.

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Black is correct statement because Grandmaster did not tie their eyes and if he brings out white hat it will be slightly visible even in dark.

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Hey, white its the right solution as its unclear that the grandmaster also wear hats (by assuming equal wining probabilities)

CarlosR

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• 2 weeks later...

Just for the record, as it seems that a number of people misunderstood this point -- it was worded ambiguously: there are four wizards total. Two losing wizards, one winning wizard, and the Grand Wizard who is the arbiter. (Although he's getting a run for his money!) Black is the only answer that would give each of the three an equal chance, for two white hats and a black hat give the man with the black hat the advantage of knowing there are no more white hats and he must have black. By the same token, if there were two black hats and one white, the persons wearing the black hats would see a white and a black hat, and have to consider the idea that his own hat could be white, in which case the person that he sees wearing the black hat would have immediately guessed the answer because he would be looking at two white hats. Since the person with the black hat is unsure, that must mean that he is seeing the same thing, a white and a black hat. Is it clear now?

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Oh, just to add this -- the winning wizard, knowing that this would be unfair, knew the Grand Wizard would have no choice but to give each one a black hat. That is the only case where keeping silent when the lights are turned on would be the natural response of each, giving them each an equal chance to guess.

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Just for the record, as it seems that a number of people misunderstood this point -- it was worded ambiguously: there are four wizards total. Two losing wizards, one winning wizard, and the Grand Wizard who is the arbiter. (Although he's getting a run for his money!) Black is the only answer that would give each of the three an equal chance, for two white hats and a black hat give the man with the black hat the advantage of knowing there are no more white hats and he must have black. By the same token, if there were two black hats and one white, the persons wearing the black hats would see a white and a black hat, and have to consider the idea that his own hat could be white, in which case the person that he sees wearing the black hat would have immediately guessed the answer because he would be looking at two white hats. Since the person with the black hat is unsure, that must mean that he is seeing the same thing, a white and a black hat. Is it clear now?

Hmmm.

3 logic masters - and 1 grand master.

Where did the wizards come from?

If they were wizards then the answer would be : he knew what colour his hat was going to be because he cast a spell to turn all five hats pink !

JSL

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Since the master that answered the Grand Master as to what color hat he was wearing before the lights went out and the hats were placed, i can only assume the the master who replied as already wering a hat and knew its color. Thats my guess.

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If 2 dudes were wearing white hats then the dude with the black hat would know he had a black hat so it wouldn't be fair. So he knew they would all have black hats to make it fair. Simple. But I don't get the question...the way I see it there's 2 dudes and the grandmaster so wheres the third dude coming from......

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The Grand Master plus three wizards. They're like his students, and he's the teacher giving them a test. The Grand Master is not participating as a test-taker, he's only there to judge. He probably wears a hat, but it's not part of the contest. It wasn't stated clearly in the original riddle. Of the three student-wizards, there is clearly one who is the star pupil. Get it?

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• 2 months later...

Because the problem is to say which color will be on your head (is on ur head), wouldn't it be best just to blurt out "Black!! White!!" very quickly?

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It is black. The solution is simple.

GM1, GM2 and GM3

First I am considering GM1 is the one who answered the question.

2 white/ 1 black

If GM1 and GM2 is wearing white GM3 would have immediately given the answer and if GM1 and GM3 is wearing white then GM2 would have responded, both GM2 and GM3 are quiet so this possibility is ruled out.

2 black/white

Case 1: GM2 looking at GM1 and GM3

if GM1 is black and GM3 is wearing white (or vice-versa) immediately GM2 would have said he is wearing black, because if GM2 is wearing white one of the other two would have immediately answered.

Case 2: GM3 looking at GM1 and GM2 - same as above

if GM1 is black and GM2 is wearing white (or vice-versa) immediately GM3 would have said he is wearing black, because if GM3 is wearing white one of the other two would have immediately answered.

Since GM2 and GM3 haven't answered yet which means GM2 (GM1 and GM3) and GM3 (GM1 and GM2) are looking at 2 black hats so for sure GM1 must be black.

So GM1 comes with an answer Black.

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• 2 weeks later...

The question is ambiguous to some extent, but at least it makes the essential point that the contestants answer by shouting out their answer. If the contestants were instead asked to write down their answers, each would have an equal chance of getting it right if all 3 hats were black, or if there were 2 black and one white hats.

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This topic is now closed to further replies.