Wizard
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June 8th, 2012 at 1:47:06 PM permalink
There is a team of three players. They are all told that each will be given a black or white hat. Each player will be able to see the hats of the other two players, but not his own. The object is to determine your own hat color, without communicating, after the hats are placed.

The host says that after the hats are placed the host will ask for a show of hands of who knows his color. These responses must be made immediately. Everyone who raised his hand must submit an answer.

If everyone who submits an answer is correct then each player will get $1,000,000. If there is at least one wrong answer, or nobody answers, they all get nothing.

The team knows this game will be played in a few hours. They are allowed to devise a strategy. What strategy should they devise?

If you figure out the answer, please put it in "spoiler" tags, PM me, or just say "I know that I know." This is the kind of thing that once you figure it out you know you're right. While I figured out an answer, I'm not sure if it is the only one that will work.

Have a nice day.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Ayecarumba
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June 8th, 2012 at 1:57:09 PM permalink
Do the players know if the other's have raised hands, or is this "communication" not allowed under the rules?
Are all white hats or all black hats a possibility?
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Wizard
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June 8th, 2012 at 1:58:41 PM permalink
Quote: Ayecarumba

Do the players know if the other's have raised hands, or is this "communication" not allowed under the rules?



They know after the show of hands only.

While any player who raises his hand must present an answer, he doesn't have to have the answer prepared at the time he raises his hand. He can base his answer on whether the other two players raise their hands.


Quote:

Are all white hats or all black hats a possibility?



Yes.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
thecesspit
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June 8th, 2012 at 2:02:01 PM permalink
Hmmm


Everyone raises their left hand if they see two different hat colours.
Everyone raises their right hand if they see hats of the same colour.

I look at contestant A. If A raises their left hand and contest B has a black hat, I must have a white hat. I can error check this by looking at B. If B raises their right hand, then contestant A should have a white hat. I'll be raising my left hand.

"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
Wizard
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June 8th, 2012 at 2:05:31 PM permalink
Quote: thecesspit

Hmmm



I read your answer, and it isn't what I'm looking for. Let me clarify a rule. You can't strategize which hand (left or right) to raise. That would violate the rule about no communicating.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
MathExtremist
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June 8th, 2012 at 2:16:32 PM permalink
Quote: Wizard

I read your answer, and it isn't what I'm looking for. Let me clarify a rule. You can't strategize which hand (left or right) to raise. That would violate the rule about no communicating.


I don't see the difference between a binary communication (hand up or not) vs. a ternary communication (left hand, right hand, neither). If one is allowed, the other should be as well -- unless you explicitly want to exclude it.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
rxwine
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June 8th, 2012 at 2:21:28 PM permalink
The one who sees two of the same color hats on the others head is the only one who should raise his hand.

Given: "Everyone who raised his hand must submit an answer"
There's no secret. Just know what you're talking about before you open your mouth.
Wizard
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June 8th, 2012 at 2:23:02 PM permalink
Quote: MathExtremist

I don't see the difference between a binary communication (hand up or not) vs. a ternary communication (left hand, right hand, neither). If one is allowed, the other should be as well -- unless you explicitly want to exclude it.



Let me put it this way. The only communication allowed is whether or not the right hand is raised. There is no other communication allowed. This would include how high the hand is raised, how long it takes to raise it, or anything like that. The answer does not lie in finding a secret form of communication.

If I wan't clear, a correct answer to the puzzle should include a hand raising strategy as well as what answer to submit for those who did.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThatDonGuy
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June 8th, 2012 at 2:24:40 PM permalink
This isn't exactly a new problem, but it did spark a bit of interest when first asked:

Only someone who sees two hats of the same color raises his hand, and answers the other color.

Of the eight possibilities, six of them are two hats of one color and one of the other; in each case, the people with the matching hats will see hats of different colors, while the person with the other color will see the two hats and guess what turns out to be the correct color. In the other two cases, all three will guess, and will guess incorrectly.

This wins 3/4 of the time. As far as I know, this is the best solution.

I actually asked a similar question on the rec.puzzles Usenet forum, and got a surprising answer when there are seven or more people involved.
Nareed
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June 8th, 2012 at 2:32:55 PM permalink
Is this a distraction from the Bob debacle?

Also, I sense the solution lies somewhere along the lines of the truthful-guard/lying-guard puzzle.
Donald Trump is a fucking criminal
Ayecarumba
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June 8th, 2012 at 2:41:55 PM permalink
Here is my proposal:

Players should only raise their hand if they see two matching colors on the other players. If all three raise their hands, the color on each is the same as the others. If only one raises their hand, the color on their head is the opposite of the two they see.
Simplicity is the ultimate sophistication - Leonardo da Vinci
thecesspit
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June 8th, 2012 at 2:45:07 PM permalink
I misread part of the instructions and then tried to niggle around the rules...


A player raises their hand if they see two different hat colours.
A player keeps their hand down if they see hats of the same colour.

I look at contestant A. If A raises their hand and contest B has a black hat, I must have a white hat.

I can error check this by looking at B. If B keeps their hand down, then contestant A should have a white hat.

I'll be raising my hand.

A - White - hand up
B - Black - hand down
Me - White - hand up

"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
Wizard
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June 8th, 2012 at 2:50:49 PM permalink
Based on discussion outside of the forum, as well as answers presented here, there seems to be another version of this puzzle than the one I just asked.

In this other version an answer must be submitted at exactly the same time as the hand is raised. The question under this other version is what strategy maximizes the chances of winning?

In my version there is a pause between the hand(s) being raised and the submitted answer. It may be that the "other" version is the way the puzzle is usually told.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThatDonGuy
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June 8th, 2012 at 2:51:25 PM permalink
Apparently, my first answer had a flaw - which I now see. I was assuming something from an earlier version of the problem that I heard.

Only those who can see two hats of the same color raise their hands.
If they see that the other two have raised their hands as well, they guess the color that they see.
If they see that nobody else's hand is raised, they guess the other color.

For example, if all three have white, all three raise their hands and guess "white".
If two are white and one is black, the two with white hats see one white and one black, and keep their hands down; the one that can see two white hats raises his hand and guesses "black".

In the "original" version, you pressed buttons (black, white, or gray for no answer), and could not see if anybody else had an answer.
Ayecarumba
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June 8th, 2012 at 2:57:54 PM permalink
Quote: Wizard

Based on discussion outside of the forum, as well as answers presented here, there seems to be another version of this puzzle than the one I just asked.

In this other version an answer must be submitted at exactly the same time as the hand is raised. The question under this other version is what strategy maximizes the chances of winning?

In my version there is a pause between the hand(s) being raised and the submitted answer. It may be that the "other" version is the way the puzzle is usually told.



Based on my previous response, here is my proposed strategy for "instant answer":
Raise hand only if you see two colors that are the same, indicating that your hat is the opposite of the two colors you see. Of the possible combinations, this strategy will only be wrong in the two instances when all three hats are the same.
Simplicity is the ultimate sophistication - Leonardo da Vinci
WongBo
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June 8th, 2012 at 2:58:25 PM permalink
delete
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rxwine
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June 8th, 2012 at 3:00:27 PM permalink
Here's my revised answer


The one who sees two of the same color hats on the others head is the only one who should raise his hand.

Given: "Everyone who raised his hand must submit an answer"

When the one person raises his hand, his hand going up knocks his own hat off and then he knows what color it is.

Kind of sounds like cheating. But I'm not sure that violated the rules.

There's no secret. Just know what you're talking about before you open your mouth.
ZPP
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June 8th, 2012 at 4:51:23 PM permalink
Each player agrees to raise his hand if, and only if, the other two hats are the same color. At least one hand will be raised because there are three hats and two colors. After the hands are raised, it's easy for each player to determine their hat color by looking at another player's hat and whether the third player's hand is raised.
P90
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June 8th, 2012 at 8:13:33 PM permalink
Each player who sees two hats of different colors raises his hand. There are 4 possible scenarios: 000, 001, 010, 011 and their inverted variants. If no hands are raised, it's 000 and all colors are the same. If one hand is raised, it's 001 or 010 and the color is opposite to that of the non-raised hand. If two hands are raised and they have the same colored hats, it's 011 and the color is opposite to both. If two hands are raised and they have different colored hats, let's hope you're all outta' gum.
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GameBoy
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June 10th, 2012 at 8:16:24 AM permalink
Solution withdrawn.
GameBoy
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June 10th, 2012 at 12:44:07 PM permalink
Solution withdrawn.
JyBrd0403
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June 16th, 2012 at 5:39:31 PM permalink
The version of this that I know goes like this. 3 people have hats placed on their heads. The hats are either white or black. Each person can say which hats they see on the other 2 people. After all three have said which hats they've seen on the other players head. They have to choose one person to give the answer of what color all the players hats are. Who should they choose so that they will win 100% of the time?
weaselman
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June 16th, 2012 at 7:16:17 PM permalink
Quote: JyBrd0403

Who should they choose so that they will win 100% of the time?


If anyone saw a white hat, choose any guy in a white hat (and he should say "white"). If nobody saw a white hat, then choose anyone, and he will guess black.
"When two people always agree one of them is unnecessary"
JyBrd0403
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June 16th, 2012 at 11:43:15 PM permalink
Quote: weaselman

If anyone saw a white hat, choose any guy in a white hat (and he should say "white").



No, the person that is chosen must say what color hat all three players are wearing. The person they choose will have enough information to correctly guess what hats all three players are wearing 100% of the time.
s2dbaker
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June 17th, 2012 at 9:35:08 AM permalink
If you see two hats of the same color on your teammates heads, then raise your hand. If they raise their hands then your hat is the same color. If they do not raise their hands then your hat is the opposite color.
Being a computer programmer, these things are pretty easy sometimes.
Someday, joor goin' to see the name of Googie Gomez in lights and joor goin' to say to joorself, "Was that her?" and then joor goin' to answer to joorself, "That was her!" But you know somethin' mister? I was always her yuss nobody knows it! - Googie Gomez
Wizard
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June 17th, 2012 at 9:47:25 AM permalink
I must be misunderstanding the problem because it seems too easy. If I'm chosen to submit the guess, for example, I already can get 2 out 3, because I can see them. It will be easy to deduce my own hat based on what just one of the others say. For example, suppose I see a black and a white hat. The person in the white hat says he sees to black hats. One of them is obvious, because I can see it too, so I must have the other one.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
JyBrd0403
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June 17th, 2012 at 10:18:44 AM permalink
You're right, it is an easy problem. I think I was in 2nd or 3rd grade when I learned this one. It's a little more difficult then. And, I'm not sure if I'm telling it right. It might be they have to choose 1 person to answer before they start, and the other 2 can say what hats they see. With that info he will be able to guess correctly 100% of the time. Reason I remember this one is I was always fascinated by the being able to not see what hat your wearing, but being able to get it right every time. It's a child's riddle, but a memorable one.
Wizard
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June 17th, 2012 at 10:36:57 AM permalink
Here is how I think you're supposed to tell it...

You are in a game with two other logicians. It is explained to each of you that each will be given a black or white hat. Each will see the other two hats, but not his own. A prize will be given the first logician to figure out his own hat color.

The hats are placed, and you see the other two logicians each have a black hat.

The host asks for a show of hands of everyone who can see at least one black hat. Everyone raises his hand.

The host says "begin."

The other two logicians do not immediately submit a guess. Let's say that a minute goes by (or enough time to figure out the answer if it were fairly obvious).

What color is your hat?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
JyBrd0403
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June 17th, 2012 at 10:46:08 AM permalink
If everyone sees at least one black hat, there's 2 people with black hats. So, if you see a black and a white hat, you have a black hat on. If you see 2 black hats, you have a white hat on.
s2dbaker
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June 17th, 2012 at 10:49:20 AM permalink
The doubt means the you are wearing a black hat.
Someday, joor goin' to see the name of Googie Gomez in lights and joor goin' to say to joorself, "Was that her?" and then joor goin' to answer to joorself, "That was her!" But you know somethin' mister? I was always her yuss nobody knows it! - Googie Gomez
ThatDonGuy
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June 17th, 2012 at 1:12:30 PM permalink
Quote: s2dbaker

The doubt means the you are wearing a black hat.


Let A and B be the other two.

If I am wearing a white hat, then B would reason immediately, "A can see a black hat, but C's hat is white, which means my hat is black." (For that matter, A would reason the same thing.) Since they don't say anything immediately, my hat must be black, so all three are black.
Wizard
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June 17th, 2012 at 2:00:31 PM permalink
I confirm the above explanation is correct.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
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