[Guest]

There are a hundred logicians who are presented with a task. They are to be placed in a line on a downward sloping hill so that each logician can see everyone in front of him. They will randomly have black or white caps placed on their heads. After this the person at the back of the line is to call out the color of his hat (which they cannot see), followed by the person in front of him and so forth.

They will not be allowed to use any tricks, such as taking different times to call out the color or any other signals of that sort. They are allowed to meet beforehand to discuss a strategy. The goal is for them, as a group, to get as many colors correct as possible.

What average percent of the colors can they get correct and how would their strategy be to obtain this maximum percentage?

For example, if each logician simply decided to randomly pick black or white, their average percentage correct would be 50%, even though the actual number of colors they correctly call could range from 0-100.

Note that it is impossible for their strategy to always get all the colors correct.

[Guest]

Could they not just each call out the colour of the person in front of them, thus guaranteeing 90% success and giving an average of 95%

There are a hundred logicians who are presented with a task. They are to be placed in a line on a downward sloping hill so that each logician can see everyone in front of him. They will randomly have black or white caps placed on their heads. After this the person at the back of the line is to call out the color of his hat (which they cannot see), followed by the person in front of him and so forth.

They will not be allowed to use any tricks, such as taking different times to call out the color or any other signals of that sort. They are allowed to meet beforehand to discuss a strategy. The goal is for them, as a group, to get as many colors correct as possible.

What average percent of the colors can they get correct and how would their strategy be to obtain this maximum percentage?

For example, if each logician simply decided to randomly pick black or white, their average percentage correct would be 50%, even though the actual number of colors they correctly call could range from 0-100.

Note that it is impossible for their strategy to always get all the colors correct.

[Guest]

Could they not just each call out the colour of the person in front of them, thus guaranteeing 90% success and giving an average of 95%

sorry that's stupid, they'd have to call out twice to do it like that

[Guest]

I think I recall another one like this: Hats on a death row!! One of my favorites puzzles! by Roolstar. The answer was something like the first person either pauses before guessing or answers straight away. If there was a pause then the colour of the person's hat in front of him is opposite to his guess, otherwise the same. After the first person there is no guess work involved.Another option would be to just say Blaaaaack or Whiiiiiiiiiite if the person in front is different to your guess else a quick Black or White if it's the same..

Edited December 17, 2008 by Braiser

[Guest]

I think I recall another one like this: Hats on a death row!! One of my favorites puzzles! by Roolstar. The answer was something like the first person either pauses before guessing or answers straight away. If there was a pause then the colour of the person's hat in front of him is opposite to his guess, otherwise the same. After the first person there is no guess work involved.Another option would be to just say Blaaaaack or Whiiiiiiiiiite if the person in front is different to your guess else a quick Black or White if it's the same..

They will not be allowed to use any tricks, such as taking different times to call out the color or any other signals of that sort.

[Guest]

The last person in line would call out the 1st, the next to last the 2nd, the second to last the 3rd and so on until the #51 person calls out the #49 person's color. Then you are sure to have 49 right and you have a 50/50 chance on the other 51 so you end up with 74.5% on average.

[HoustonHokie]

I agree this has been done before, but this is the answer I like best (it feels less like cheating )

On average, they should be able to guess 99.5 of the 100 caps correctly. The first guesser will see an odd number of one color and an even number of the other. By agreement of the group, he calls out the color he sees an odd number of times in front of him. Half of the time he's right, half of the time he's wrong on the color of his own cap, but he has provided a valuable clue for everyone else. Let's say he saw an odd number of blacks and therefore guessed black. The second guesser counts the caps he can see and finds either they are both even or they are both odd. If they are both odd, he knows he's wearing a white cap; otherwise, he's wearing black. He calls out his correct color. Each successive logician keeps track of whether each color should be odd or even, and by comparing what he sees with his mental tally, he can guess the correct color of his own cap. Only the first logician could call an incorrect color, and he'll be right half the time. So, on average, they will guess 99.5 of the caps correctly.

[Guest]

I agree this has been done before, but this is the answer I like best (it feels less like cheating )

On average, they should be able to guess 99.5 of the 100 caps correctly. The first guesser will see an odd number of one color and an even number of the other. By agreement of the group, he calls out the color he sees an odd number of times in front of him. Half of the time he's right, half of the time he's wrong on the color of his own cap, but he has provided a valuable clue for everyone else. Let's say he saw an odd number of blacks and therefore guessed black. The second guesser counts the caps he can see and finds either they are both even or they are both odd. If they are both odd, he knows he's wearing a white cap; otherwise, he's wearing black. He calls out his correct color. Each successive logician keeps track of whether each color should be odd or even, and by comparing what he sees with his mental tally, he can guess the correct color of his own cap. Only the first logician could call an incorrect color, and he'll be right half the time. So, on average, they will guess 99.5 of the caps correctly.

Correct. 99.5% is the best you can do.

[bonanova]

Repeat post. Locked.