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.
Are all white hats or all black hats a possibility?
Quote: AyecarumbaDo 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.
Quote:Are all white hats or all black hats a possibility?
Yes.
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.
Quote: thecesspitHmmm
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.
Quote: WizardI 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.
Given: "Everyone who raised his hand must submit an answer"
Quote: MathExtremistI 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.
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.
Also, I sense the solution lies somewhere along the lines of the truthful-guard/lying-guard puzzle.
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
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.
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.
Quote: WizardBased 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":
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.
Quote: JyBrd0403Who 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.
Quote: weaselmanIf 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.
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?
Quote: s2dbakerThe doubt means the you are wearing a black hat.
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.