[Monte314]

A starship beams a group of settlers down to an uninhabited planet. 100 of them have exactly two brown eyes, and 100 of them have exactly two blue eyes.

Strangely, though, none knows his own eye color. Right after the starship has beamed the settlers down, it sends them all this message: "At least one of you has blue eyes." The settlers are all excellent thinkers -- if a conclusion can be deduced logically from the available information, each one will do it almost instantly.

Every night at midnight, the star ship passes overhead, quickly performs a remote mind-scan on the settlers, and instantly beams up anyone who has managed to logically deduce their own eye color. The remaining settlers all attend a community breakfast each morning, but they cannot communicate in any way. However, since they can see each other at breakfast, each knows how many of each eye color is present (excluding themeselves, of course). All the settlers know all the facts in this paragraph.

Because the settlers don't know their own eye color, they do not know the number of people having each eye color, nor do they know for certain all the colors that are represented. Any given blue-eyed person will see 100 people with brown eyes and 99 people with blue eyes, but that does not tell him his own eye color. As far as he knows the color totals could be 101 brown and 99 blue... or 100 brown, 99 blue, and he himself could have green eyes!

Question: Who is beamed up, and on what night(s)?

NB: There are no mirrors or reflecting surfaces of any kind. This is not a "trick" question; the answer is based entirely on logical reasoning. It doesn't depend on subtle wording or anyone lying or guessing, and it doesn't involve people doing something silly like creating a sign language or doing genetics.

And, lastly, the answer is not "no one leaves."

[thod]

Suppose there was one blue eyed man. He would see all the others as brown thus indicating that he was blue and he would be beamed up at midnight.

If there were two, then each would see one other blue eyed man. If that blue eyed man was the only one he would go at midnight. So if he is still around after that it means he was unable to determine it. This indicates you too are blue eyed and you both get beamed up at midnight the on the second day.

So how about you see two blue eyes, they didn't get beamed up on the second day, that tells you that you are blue number three.

You can continue like this, so nothing happens and then on the the 100th day all the blue eyed people are beamed up. Seeing them all go tells the rest they are brown and so go up on the 101st day.

"Fie upon this quiet life! I want work."

[Monte314]

Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

Suppose there was one blue eyed man. He would see all the others as brown thus indicating that he was blue and he would be beamed up at midnight. If there were two, then each would see one other blue eyed man. If that blue eyed man was the only one he would go at midnight. So if he is still around after that it means he was unable to determine it. This indicates you too are blue eyed and you both get beamed up at midnight the on the second day. So how about you see two blue eyes, they didn't get beamed up on the second day, that tells you that you are blue number three. You can continue like this, so nothing happens and then on the the 100th day all the blue eyed people are beamed up. Seeing them all go tells the rest they are brown and so go up on the 101st day. "Fie upon this quiet life! I want work."

Dang! I thought this problem was hard, but I didn't consider the "Thod Factor".

(note: though technically, by the construction of the problem all the brown-eyed people know is that their eyes are not blue... could be green... so they are stranded forever.)


This thread is over!

[cal]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	(note: though technically, by the construction of the problem all the brown-eyed people know is that their eyes are not blue... could be green... so they are stranded forever.)

So that no one hopefully gets left behind: Maybe I missed something in your 1st post, Monte, but it didn't seem to suggest in it that the settlers couldn't take a logically based guess as to their own eye colour. If they were as you said in your 1st post - "The settlers are all excellent thinkers -- if a conclusion can be deduced logically from the available information, each one will do it almost instantly" - then could it be justifiably assumed that they would look around, notice that only 2 eye colours were present, and deduce that the odds would be good that they had eyes that were one of the colours. If they guessed one colour the 1st night and weren't beamed out, then they could guess the other colour the second night. If they were still there after that, then they'd have to start assuming their eyes were of a colour different from all the other settlers. But since there wasn't a 3rd colour, all should be out of there by the 2nd night.

[Monte314]

Originally Posted by cal To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

So that no one hopefully gets left behind: Maybe I missed something in your 1st post, Monte, but it didn't seem to suggest in it that the settlers couldn't take a logically based guess as to their own eye colour. If they were as you said in your 1st post - "The settlers are all excellent thinkers -- if a conclusion can be deduced logically from the available information, each one will do it almost instantly" - then could it be justifiably assumed that they would look around, notice that only 2 eye colours were present, and deduce that the odds would be good that they had eyes that were one of the colours. If they guessed one colour the 1st night and weren't beamed out, then they could guess the other colour the second night. If they were still there after that, then they'd have to start assuming their eyes were of a colour different from all the other settlers. But since there wasn't a 3rd colour, all should be out of there by the 2nd night.

Excellent suggestion. I was trying to prevent such "mental experimentation" by requiring that the inhabitants logically deduce their eye color, not just have a working hypothesis. But, it sounds like I need to work on the wording a little bit!

Thanks!

[cal]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Excellent suggestion. I was trying to prevent such "mental experimentation" by requiring that the inhabitants logically deduce their eye color, not just have a working hypothesis. But, it sounds like I need to work on the wording a little bit! Thanks!

So I 'cheated'?
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Your qualifying 'Beam me up, Scotty!' comment in your 1st post - "Every night at midnight, the star ship passes overhead, quickly performs a remote mind-scan on the settlers, and instantly beams up anyone who has managed to logically deduce their own eye color" - reminded me of a quote about logic and its use in 'real' situations: "The purpose of logic should not be so much to find the final conclusion, but to make sure that it is sound once it has been found. There is no doubt that this proof should be as precise and rigorous as possible. But this sort of proof can be just as easily applied to a conclusion that has been reached by lateral thinking as to one that has been reached by vertical thinking".

To me, it seemed logical to use 'logic' here in whatever 'lateral thinking' way that seemed to fulfill the criteria sufficiently, and could lead to a satisfactory result.

[Monte314]

No, I don't think you cheated! I think you pointed out a weakness in my presentation of the problem.

None of the problems I am interested in is a lateral thinking problem; they are about creative, but direct, application of mathematical techniques.

...and, I really do very much appreciate your comments.

[cal]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	No, I don't think you cheated!

I know. I was just kidding. Hence the
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
face. (You seem too intelligent and too nice a guy for me to have believed you would've suggested/implied such in this instance.)

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	...and, I really do very much appreciate your comments.

Thanks, Monte.

[Arcani]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Excellent suggestion. I was trying to prevent such "mental experimentation" by requiring that the inhabitants logically deduce their eye color, not just have a working hypothesis. But, it sounds like I need to work on the wording a little bit! Thanks!

Just include that the settler must be certain about their eye color, there can be no doubt in his/her mind, thus eliminating random guessing.

This one's a classic, I think Thod probably solved it before, I know I have.

I'm really enjoying these, please keep them coming.