Interesting Logic Problem

[BeanDip4All]

Tonight in class my professor threw out a bonus question with an interesting caviat: If we answer correctly, he takes us out to dinner here. (yum!)

I've already written up and submitted my proof, but I thought with tomorrow being Friday and all, people might want to take a stab at discrete math instead of looking at NSFW pix. So if anyone wants to have play, here you go. I won't throw down for expensive sushi but think of the accolades you'll get here for being an Interweb Mathy Dweeb.

Several people with assorted eye colors live on an island. They are all logicians -- if a conclusion can be logically deduced, they will act on it instantly. No one knows the color of their own eyes. Each night at midnight, a ferry comes to the island. If anyone has figured out what color their own eyes are, they will/must leave the island that midnight. Everyone sees everyone else at all times and does keep count of the number of people they see with each specific eye color (excluding themselves), but cannot otherwise communicate. Everyone on the island knows the rules in this paragraph.

On the island there are 100 people with blue eyes, 100 people with brown eyes, and the Doctor (who happens to have green eyes). Any random person with blue eyes can see 100 brown eyed people and 99 blue eyed people (the Doctor, with green eyes), but that does not tell them about their own eye color; to their knowledge the totals could be 101 blue and 99 brown. Or 100 blue, 99 brown, and he could have purple eyes.

The Doctor is allowed to speak once (let's say at midnight), on one day in all their countless years stuck on the island. When he does this, he says the following:

"I see someone with blue eyes."

Who should the island, and on what specific night?

The island has no reflecting surfaces whatsoever, no mirrors, etc. This is not a trick question and you can logically find the answer. The answer can be derivine from math and things like creating a sign language or genetics will not be accepted. The Doctor is not making direct eye contact with anyone in particular. Rather, he is simply stating, "There is at least one blue-eyed person on this island who is not me."

Finally, the answer is not "No one leaves the island."

[Squatch]

the doctor leaves the island

[Steven S. Dallas]

The plane falls off the treadmill, gets sucked underneath it, and explodes, killing hundreds.

[Viva]

I thought with tomorrow being Friday and all, people might want to take a stab at discrete math instead of looking at NSFW pix.

JONG!!!!!

[soulman978]

the doctor leaves the island

That's exactly what I was thinking too.

[smitchell333]

This is the question?:
"Who should the island, and on what specific night?"

The answer is - This is not a coherent question and therefore there is no answer.

Salmon Cone-style please

[AbsolutStoli]

the ferry pilot leaves the island, every night.

[irul&ublo]

the ferry pilot leaves the island, every night.

A ferry pilot? Tim Hardaway will not fly on that plane.

[f2f]

nobody leaves for the first 99 midnights, then all the blue eyed people leave at once.

once the phrase is uttered a blue eyed person needs to wait 99 midnights before they can make the decision (because each of the other 99 people who have blue eyes can leave any midnight). after 99 days a blue-eyed person knows that every other blue-eyed person sees at least 99 other blue-eyed people, so s/he leaves. a brown-eyed person has to wait a day more, because they have 100 blue-eyed observations, but at that time all 100 blue-eyed people are left.

[MeatPuppet]

The Doctor is allowed to speak once (let's say at midnight), on one day in all their countless years stuck on the island.

The fact that the time of day the Doctor speaks, as well as the fact that the ferry comes once a day, is included in the question implies that the daily cycle has some significance. If the Doctor only spoke once, the daily cycle would be meaningless. So....

If he speaks only one time, then I'm with AbsolutStoli, ferry pilot. But if the Doctor speaks once a day every day, then there is another answer...but that's not how the question was worded.

[pappag]

Doctor's statement is meaningless. I think it's put there to throw people off and get you to think about something you needn't spend time thinking about.

Start with 10 people. A blue-eyed person sees 4 other blues and 5 browns. Browns see 5 blues and 4 browns. First midnight - no one leaves because no one is 100% certain. 2nd, 3rd, and 4th midnights, no one leaves because, again, no one is 100% sure.

The logic now inside the blue-eyed mind is that 4 nights have passed and no one has left. I see only 4 blue eyes and 5 browns. If I was brown (or any non-blue color) that means all the blue-eyed people would have seen 3 blues and 6 browns and they would all have left by now. But no one has left yet, so that means I must have blue eyes, and therefore I can leave now. The remaining 4 blue-eyed people think the same thing and leave together on the 5th night.






As f2f alluded, in this particular problem, all the blue-eyed people leave on the 100th midnight:

Each blue-eyed person sees 100 people with brown eyes and 99 with blue eyes.
Each blue-eyed person assumes that they have non-blue eyes.
Going further, all the blue-eyed people that each blue-eyed person recognizes sees 98 people with blue eyes.
To each blue-eyed person, there are only 99 people with blue eyes.

When the 99th midnight comes and goes, no one leaves the island because there are in fact 100 people with blue eyes and not 99 (as was the assumption). Each blue-eyed person follows the same deduction and leave together on the 100th midnight.

[RootSkier]

Oh fuck, this is like trying to get the perverted minister across the ferry, only he can't be left alone with the kids or the wife, and even after we learned the answer, I still couldn't do it.

[MeatPuppet]

As f2f alluded, in this particular problem, all the blue-eyed people leave on the 100th midnight.

Assuming the question is worded incorrectly and the missing information supplied, wouldn't this^^^assume that the Doctor is seeing a different blue eyed person every time he speaks?

[f2f]

your prof is trying to teach you that a logic system is not necessarily solvable this instant but can take stages. actually Nick is wrong in saying that the doctor's statement is useless, since had the doctor not said anything about a blue-eyed person nobody could be sure what their eye color was (i.e. everybody could think they have purple eyes). the doctor's statement is the input in the logic system (the zero or one down the cpu pipeline if you wish).

to simplify further, start with one blue-eyed person (the number of people with brown eyes is irrelevant here since they always see at least one blue-eyed person more than necessary): after the doctor says there is a blue-eyed person then the person with blue eyes, which sees no other person with blue eyes, can deduce that they are the one with the blue eyes.

if two people have blue eyes, then they both see a person with blue eyes, so they can't take action. one midnight passes, then each person with blue eyes knows that the other has seen at least one more person with blue eyes. since everybody else is brown-eyed, then that person must be them.

you can continue by induction.

[InspectorGadget]

I'll take NSFW for $200 Alex.

[AbsolutStoli]

another solution is that all blue eyed people leave the island the midnight that the doctor makes the announcement. logically, upon hearing that there is at least one blue-eyed person on the island, everyone will consider the possibility that they are that one person, and will instantly act on that logic (as per the rules). all of the people will say to the ferry pilot that they have blue eyes and the 100 that actually do will leave. the brown-eyed people will not, since they did not get their eye color right, but they will know that they don't have blue eyes. but they will have to wait for however long before the dr. makes another announcement as to what color eyes he sees.

[Sirshredalot]

I think f2f is 100% on the right track, but I am too drunk to turn that into a formal proof.

[Arty50]

If he speaks only one time, then I'm with AbsolutStoli, ferry pilot. But if the Doctor speaks once a day every day, then there is another answer....

[Doctor rolls up to every hottie on the island] "Do you know your eyes are the color of my Porsche?"

[bklyn]

The plane falls off the treadmill, gets sucked underneath it, and explodes, killing hundreds.

I'll take NSFW for $200 Alex.

[Doctor rolls up to every hottie on the island] "Do you know your eyes are the color of my Porsche?"

You guys crack me up.

[snow_slider]

I prefer boobies to logic.

[Tippster]

Kill them all, let God sort them out.

[smalls]

If we answer correctly, he takes us out to dinner here. (yum!)

$600 cucumber roll?? holy crap.
that cucumber better pleasure my hardy boys and clean my house for that kind of coin.

[MeatPuppet]

[Doctor rolls up to every hottie on the island] "Do you know your eyes are the color of my Porsche?"

He would never have to worry about making breakfast the next morning.

[summit]

after the doctor says there is a blue-eyed person then the person with blue eyes, which sees no other person with blue eyes, can deduce that they are the one with the blue eyes.

There is no context given to the Doctors statment. If the problem stated how many people were in his presence when he made his one and only statement, then it could have some weight.

Each person knows EITHER:
Blue>=100
Brown>=99
Green>=1

OR

Blue>=99
Brown>=100
Green>=1


The doctor uttering the statement "Blue>=1" one time only ever is ABSOFUCKINGLUTELY MEANINGLESS until you are given context to his statement... how many people of what eye colors are present when he makes that observation... assuming his comment was even time specific...

A general statement of "Blue>=1" adds nothing to anyones knowledge because everyone already knew that because eveeryone already knows "Blue>=99" or "Blue>=100"

The only real wiggle room in this problem is that there are no reflecting surfaces on the *island* They didn't say anything about the people... eyes are reflective surfaces ... but I don't think he wants that answer. Just kick the prof in the nuts and tell him to eat a bucket of dicks.

logically, upon hearing that there is at least one blue-eyed person on the island, everyone will consider the possibility that they are that one person, and will instantly act on that logic

How is that logical? They know there are other blue eyed people. They don't act on possibilities, only logical certainties.

Assuming the question is worded incorrectly and the missing information supplied, wouldn't this^^^assume that the Doctor is seeing a different blue eyed person every time he speaks?

And its flat wrong b/c the doc only speaks once... ever... proving that this is a fucking retarded island to live on and they should all die. Also, the question is retarded and you should kick your instructor square in the nuts.

[pappag]

actually Nick is wrong in saying that the doctor's statement is useless, since had the doctor not said anything about a blue-eyed person nobody could be sure what their eye color was (i.e. everybody could think they have purple eyes). the doctor's statement is the input in the logic system (the zero or one down the cpu pipeline if you wish).

But the doctor is saying something that everyone knows. Yes, the doctor sees blue eyes, but so does everyone else. If everyone can see at least one blue-eyed person (which is the case with all blue-eyed seeing 99 blue-eyed people and all brown-eyed seeing 100 blue-eyed people), then the doctor's statement is completely redundant. He is stating something that is effectively common knowledge amongst the 200 people trying to get off the island.

Perhaps in the mathematical sense, the doctor's statement has validity, but I just don't see it.





And for clarity, the statement causing confusion should read "Who should leave the island, and on what specific night?"