Nice brainteaser

[partygirluk]

100 male utilitarian logicians are told that tomorrow morning they are to be lined up and a hat placed on their head. Meanwhile they can communicate with each other. The hat will be either black or white with the probability of black for each individual 2/3 (edited to clarify: Each logician has a 2/3 chance of getting a black hat independent of what colour hat the other logicians have).

They will be asked from the backmost logician forwards what colour their hat is. If they guess correctly they are freed, if they get it wrong they die. Each man can only see the colour of the hat of the men in front of him, but they known what has happened beforehand (i.e. they know what each guy has guessed and whether they were freed or killed).

What is the best strategy for the logicians to devise in order to have the highest EV of saved lives? What about the highest number of guaranteed saved lives?

If you have heard this question (and answer) before please do not list the solution in your post!

[tbach24]

Obviously they should just guess black until the odds of black/white change into the favor of white, if they ever do.

[Macdaddy Warsaw]

Wouldn't everyone guessing black guarantee the most lives?

And wouldn't saying black until 33 people who had said black had survived be the most +LV value? EDIT: 33 people who said black and lived or said white and died but nobody should be saying white until this point anyway.

I have no idea.

[gaming_mouse]

If the original decision of black is just 2/3, and all the decisions are independent, then it makes no difference at all (EDIT: no difference what other people have done). If you are saying that only 2/3 of this logicians can have black hats, then how many have black hats, 66 or 67?

[1800GAMBLER]

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until the odds of black/white change into the favor of white, if they ever do.

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The hat will be either black or white with the probability of black for each _individual_ 2/3

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Unless i'm being stupid, which i must be because otherwise this seems like a dumb brainteaser; 2/3 chance black here does not meean there are 67 black hats and 33 white hats. It means if the people giving out the hats start off using 2 black and 1 white and the first person gets a white hat, they then replace the white hat and move on to the next guy.

Meaning the best strategy would be for everyone to pick black and 2/3s will live.

[Duke]

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Obviously they should just guess black until the odds of black/white change into the favor of white, if they ever do.

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You're obviously wrong.

I heard a similar one before so I won't answer.

~D

[Thythe]

To me it sounds like the probability is always 2/3 for a black hat. In this case then it is possible for all 100 hats to be black or for all 100 hats to be white, though either case is unlikely. Is this correct or is there an exact amount of each color hat (i.e. 33 white 67 black)?

[jason_t]

After you posed this to me last night, I thought of a rather interesting variant.

We place no assumption on the distribution of the hats. Instead of the logicians being in a line, they are all in a room and can see every hat but their own. They do not submit answers verbally. They do not know the results of anyone else's guess. So, the only information that each logician has is the colour of every other logician's hat but their own.

Same questions. What is the best strategy for the logicians to devise in order to have the highest EV of saved lives? What about the highest number of guaranteed saved lives?

The answer is surprising.

[tbach24]

[ QUOTE ]
[ QUOTE ]
until the odds of black/white change into the favor of white, if they ever do.

[/ QUOTE ]

[ QUOTE ]
The hat will be either black or white with the probability of black for each _individual_ 2/3

[/ QUOTE ]

Unless i'm being stupid, which i must be because otherwise this seems like a dumb brainteaser; 2/3 chance black here does not meean there are 67 black hats and 33 white hats. It means if the people giving out the hats start off using 2 black and 1 white and the first person gets a white hat, they then replace the white hat and move on to the next guy.

Meaning the best strategy would be for everyone to pick black and 2/3s will live.

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They can see the 100th one ahead of them so that they can adjust their count. Also, they know who the count of who is dead/alive behind them and their guesses, so they can adjust their guesses.

[Duke]

Why'd you even include the 2/3 thing. That's getting everyone thinking about it incorrectly.

~D