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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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Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re: Logicians don't discuss strategy (spoiler?), but | Comment 2 of 6 |
(In reply to Logicians don't discuss strategy (spoiler?) by Steve Herman)

 Logicians don't discuss strategy (spoiler?),

I never saw any "official" solution and my 1st approximation
was exactly like yours:   "if one sees two equal colors, write down the opposite - otherwise offer a "don't know", - the team will win 75% of the rounds played.
I also agree that no prior coordination is needed if three logicians were participating.

My next improvement , needing prior agreement  between 3
people (not necessarily logicians!)provides 100% rate of success as long as the conditions of the process are as described.

One more hint : If you, Steve, your friend and I were the team playing for money, we would accept odds like 1:2.8 (instead of 1:3)  and willingly introduce now and then errors - preventing the disclosure that our strategy cannot fail.

Your turn - 

  Posted by Ady TZIDON on 2013-12-20 11:00:36
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