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Hats revisited (Posted on 2013-12-19) Difficulty: 2 of 5
Three people are trying to win the following game as a team:

Each of them is put on a hat of either red or blue. Assume an equal chance of getting a red or a blue hat, separately for each participants.
Each one can only see the other people's hats, but not his own.
He has to guess the color of his own hat by writing down either "Red", "Blue", or "Don't know".
After all three people submit in writing their guesses, they would jointly win if:

1. At least one of them guessed right,
2. None of them guessed wrong .

"Guessed right" is defined as guessing a color that is the color of the hat.
"Guessed wrong" is defined as guessing a color that is NOT the color of the hat.
It's neither "right" nor "wrong" if "don't know" is submitted as an answer.

Those three people can establish a joint strategy before the hats are put on their heads.
After the hats are on, they can neither communicate to each other nor see other guesses.

What strategy would give them the best chance of winning and what's the probability of winning under that strategy?

Source: Allegedly posted in the elevator of UC Berkeley Math department.

No Solution Yet Submitted by Ady TZIDON    
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re(2): Logicians don't discuss strategy (spoiler?), but | Comment 3 of 6 |
(In reply to re: Logicians don't discuss strategy (spoiler?), but by Ady TZIDON)

Well, if you need 100%, then I think some signalling is called for. 

Here's a simple approach.  Ady, my friend and I are playing.  I will write "don't know", but I will do it as soon as possible if I see Ady's hat is red, and I will delay a minute if it is blue.  Ady will write Blue or Red, depending on whether or not I delay.  My friend, who is not aware of the method, has been instructed to always write "don't know". 

Or, if we want to show off,  I will write "don't know", but I will do it as soon as possible if I see two matching hats, and I will delay a minute if I see two different colors.  Ady will deduce his hat color based on my friend's hat color and my delay (or lack thereof).  My friend does the same, based on Ady's hat color and my delay.  So we wind up with two correct answers.

  Posted by Steve Herman on 2013-12-20 18:44:27
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