Suppose you and 11 friends are invited to play a game. The game is as follows:

1) All of the 12 participants are blinded folded and have either a red or blue hat placed on each of their heads.

2) The host then randomly arrange them in a circle in such a way that each participant can only see the 4 neighbors immediately to his/her left and the 4 neighbors immediately to his/her right.

3) The blindfolds are removed, and each participant can look at the hat of his 4*2 = 8 immediate neighbors.

4) Each person must then write down a guess for his/her hat. Each guess must either be 'red' or 'blue', and must be written at the same time.

5) The game host then looks at all the guesses. If they are ALL correct, the 12 participants win. If 1 or more guesses are incorrect, the participants lose.

Please assume that the players do not cheat. Any attempt to exchange information through words, utterances, signs, facial expressions, delays in writing their answer, etc. will lead to an automatic loss. If everybody randomly guess their hat, then the chance of winning the game is 1/2^{12} = 1/4096. Fortunately, there are better strategies than that.

The players can discuss a strategy before playing the game. Determine a strategy that has a winning rate equal to or greater than 3/64 (that's 192 times better than random guessing).

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## bushindo 14

Suppose you and 11 friends are invited to play a game. The game is as follows:

1) All of the 12 participants are blinded folded and have either a red or blue hat placed on each of their heads.

2) The host then randomly arrange them in a circle in such a way that each participant can only see the 4 neighbors immediately to his/her left and the 4 neighbors immediately to his/her right.

3) The blindfolds are removed, and each participant can look at the hat of his 4*2 = 8 immediate neighbors.

4) Each person must then write down a guess for his/her hat. Each guess must either be 'red' or 'blue', and must be written at the same time.

5) The game host then looks at all the guesses. If they are ALL correct, the 12 participants win. If 1 or more guesses are incorrect, the participants lose.

Please assume that the players do not cheat. Any attempt to exchange information through words, utterances, signs, facial expressions, delays in writing their answer, etc. will lead to an automatic loss. If everybody randomly guess their hat, then the chance of winning the game is 1/2

^{12}= 1/4096. Fortunately, there are better strategies than that.The players can discuss a strategy before playing the game. Determine a strategy that has a winning rate equal to or greater than 3/64 (that's 192 times better than random guessing).

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