Suppose you and 4 friends are invited to play a game. The game is a variation of the hat puzzle with monetary incentive. The game is as follows:

1) You and your 4 friends each pay 10 dollars to participate in the game (50 dollars total).

2) All participant is then blind folded and have either a red or blue hat placed on each of their heads.

3) The host then randomly arrange them in a circle in such a way that each participant can only see the 1 neighbor immediately to his/her left and the 1 neighbor immediately to his/her right.

4) The blindfolds are removed, and each participant can look at the hat of his two immediate neighbors.

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

6) The game host then looks at all the guesses. If they are ALL correct, the 5 participants win a sum of 1000 dollars (20 times the money required to play the game). If one or more guesses are incorrect, the participants lose the 50 dollars.

7) Any attempt to cheat ( trying to exchange information through words, utterances, signs, facial expressions, delays in writing their answer, etc.) will lead to disqualification and forfeiture of the 50 dollars.

Obviously, if you and your 4 friends each randomly guess, your change of winning is 1/2^{5} = 1/32. Over about 30 such games, your group will shell out 1500 dollars while winning an expected 1000 dollars only. Fortunately, there are better strategies than random guessing. Determine a strategy that will give you and your friends positive expected winnings in the long run. (Bonus) What is the strategy with the highest expected winning?

## Question

## bushindo 14

Suppose you and 4 friends are invited to play a game. The game is a variation of the hat puzzle with monetary incentive. The game is as follows:

1) You and your 4 friends each pay 10 dollars to participate in the game (50 dollars total).

2) All participant is then blind folded and have either a red or blue hat placed on each of their heads.

3) The host then randomly arrange them in a circle in such a way that each participant can only see the 1 neighbor immediately to his/her left and the 1 neighbor immediately to his/her right.

4) The blindfolds are removed, and each participant can look at the hat of his two immediate neighbors.

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

6) The game host then looks at all the guesses. If they are ALL correct, the 5 participants win a sum of 1000 dollars (20 times the money required to play the game). If one or more guesses are incorrect, the participants lose the 50 dollars.

7) Any attempt to cheat ( trying to exchange information through words, utterances, signs, facial expressions, delays in writing their answer, etc.) will lead to disqualification and forfeiture of the 50 dollars.

Obviously, if you and your 4 friends each randomly guess, your change of winning is 1/2

^{5}= 1/32. Over about 30 such games, your group will shell out 1500 dollars while winning an expected 1000 dollars only. Fortunately, there are better strategies than random guessing. Determine a strategy that will give you and your friendspositiveexpected winnings in the long run. (Bonus) What is the strategy with the highest expected winning?Edited by bushindo## Link to post

## Share on other sites

## 10 answers to this question

## Recommended Posts

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.