Atheist, Can I test your logic?

Question

You come here to test our faith, and since most of you claim that you use your logic to form your beliefs...can I test your logic? Please don't cheat and look at answers.

Solve:
There's an isolated island with k natives living on it. The natives cannot communicate with each other in any way. All the natives have blue eyes. One night, the devil comes, and turns n > 0 of these natives from blue eyes to having brown eyes. All natives know that the devil has turned at least one of them from blue eyes to brown eyes. Each night the natives gather in a circle and look at each other. However, if a native has deduced that he/she has brown eyes, they don't come to any future gatherings.

Do any of the natives not come to future gatherings? If so, how long does it take until all n of the brown-eyed don't come to future gatherings? (NOTE: There are no mirrors nor any reflecting surfaces of any kind on this island, so the natives are unable to see the color of their own eyes.)

Please show your work as in any good proof :)

This is basic logic using induction, and there is enough information to solve the problem. Not one serious attempt to solve yet? What does that tell me about your logic abilities?

So I think I can safely assume that none of you have been educated in logic. But you all claim to use it to get the answer to human kinds biggest question. This is funny.

Chippy, this is a simple logic question that could be asked in any college across the nation. Are all math problems rational, the classic a train traveling x mph in one direction, another in the opposite direction at y mph...when will they collide...do you dodge that question as well by asking why they are on the same track? If you would like, I can rephrase the question without including the devil and I can make them meet in the daytime. Nothing will change though, because you obviously do not know how to solve the problem.

Chippy, I see that I have upset you. Sorry you are not as solid in logic as you thought you were.

This has revelance to beliefs because atheist come here all the time saying they used their "logic" to deduce that there is no God. So, I just want to see if I can take any of them seriously, or if they are all sheep following one another.

yay for salient2 and Bad Liberal.

Bad Liberal, I really don't care if you think I can solve it or not.

Here is the thing, there are atheist on here everyday claiming they have superior logic reasoning because they have logically concluded that no God exist. But none of them can complete a simple logic proof. What does that tell you? Most of them tried to dodge or criticize the question because they had no idea where to start. Isn't that what they accuse believers of doing?

Kjelstad, Andymcj78 (atheist), Magley64, Vishal , Fu Quan , vinslave , Jeff S, Jedi Master , Edge , joshmiss, icarus62 , Jiraiya, Chippy, Rev Ron, Swag, when2foldem, lilith , haysoos2, mad dog, and Lucid Interrogator...thanks for playing and now I know none of you have much knowledge of logic...so it will be hard to take you seriously.

I'm not trying to "prove" God, I am trying to prove that most atheist here that claim they use logic to form their beliefs have little understanding of logic in the first place.

salient2, I'm not comparing believers to atheist. As far as I know, not many believers argue logic for their belief in God. But most atheist do argue their use of logic for KNOWING there is no God. How can they come to this conclusion when most of them can't apply basic logic. They attack the question saying that since I included the "devil" in the question, then it is not valid...just dodging the question because of their lack of knowledge to solve it.

Vishal, I will take you off the list...but you didn't solve until after salient2 did. J/K. You seem to have a good grasp of the inductive proof.

Fu Quan, Vishals logic is exactly correct. Everyone will attend the meeting until the n'th night, then on the n+1 night only the blue eyed natives will come to the meeting. salient2, explains it a little more clearly, see his for a more detailed explanation.

Answer 1

If there were exactly one brown-eyed native . He would see that everyone else has blue eyes. The second night the brown eyed person would not come.

Now suppose there were two brown-eyed natives. Each of them would see one other person with brown eyes, so neither would leave on the second night. However seeing that the only other brown-eyed native had stayed on the second night, each native could then conclude that he had brown eyes, and therefore all brown eyed natives would not come on the third night.

Proof By Induction: We see that if n < 3 then all brown eyed natives will leave on the n+1th night. We assume this is true for all values less than n and try to prove it for n.

If there were n natives with brown eyes then on the n+1th night each of the n brown-eyed natives, seeing n-1 other brown-eyed natives who had showed up the previous night, would correctly deduce that he has brown eyes, so each of the brown eyed natives would not come on the n+1th night.

This is because they would realize that if they are not brown-eyed there would be n-1 brown eyed people and that knowing the induction hypothesis all the brown eyed people should have not shown up on the nth night.
The fact this did not happen shows that they have brown eyes. Therefore they ( and all their brown eyed brethren ) will not show up on the n+1th night.

Perhaps the fact I am a mathematician has more to do with my ability to solve this than the fact I am an atheist. Not all of us atheists are highly skilled at math and logic. Of course you need to ask a similar question of Christians to get a control sample.

The fact many of my atheist brethren are unable to solve this has nothing to do with whether theism is correct. I suspect most of them would be unable to follow a detailed explaination of exactly why I am an tegmarkian neoplatonist atheist, but why does that matter? Most of them on the other hand are fully capable of seeing through the highly illogical reasons typical Christian Creationists use for why their god exists.

Answer 2

Well, first of all, is there a camp fire involved so that they can actually see the color of each other's eyes? (you said they gathered at night). I will assume there is I suppose...

I had to reread the question - I did not realize that they
knew that some would have brown eyes - so I will start
over again!

I have to assume that they can't even communicate
by pointing or even looking surprised at those with
brown eyes! I also have to assume that they can
see color!

I also have to say that k > 1 since you used the plural
form of natives and they looked at one another.

If they know that at least one will have brown eyes, then
they will expect at least one, so, naturally, if they see
no one with brown eyes then they know they are the
only one with brown, so they do not come back to
the gathering.

Of course, I don't know how they even know to gather
since they don't communicate - just tradition I suppose!

If 2 have brown eyes then both will return for the
next meeting. I already saw the answer explaining that
on the third meeting, the two would have figured it out,
so I don't take credit for that!

Of course, if there were only 2 natives to begin with,
then there would not be a gathering at all. He He.

What if, say, 5 have brown eyes and keep returning.
So, I have brown eyes and keep seeing 4 others with
brown eyes. Would I ever know if I had brown eyes?

It is the same if all had brown eyes and there were
more than two natives.

(sorry I did not read it correctly at first)
I had the same basic ideas, though - I just
left out the fact that they knew some eyes
would be brown - so I deleted and started over! :0) -
it goes to show you that distractions are
not good to read logic questions by! He He

I don't know what this has to do with faith.
I am a Christian.
I know we can't ever know ourselves spiritually
without Christ.
We don't know that we are spiritually dead or alive,
weak or strong - God knows us better than we
know ourselves!

I am inerested at what the purpose and answer to
this question is!

Answer 3

First, since I've yet to see an inductive argument for god, unfamiliarity with inductive proofs is not an indictment against the arguments of atheists. the only indictment would be if you found a logic error with the specific argument.

The fact that few are able to solve this does nothing to produce evidence in favor of a god.

Still, you asked the question, here is the answer:
------------
Case n = 1
meeting m = 1, k natives show up.
single native sees all others with blue eyes.
All other natives see 1 with brown eyes
Concludes he is the single one with brown.
does not return

meeting m = 2, k-1 native show up
since single brown eyed native did not return he must have seen all blue. Everyone left is blue eyed.

case n = 2
meeting m = 1, k natives show up
2 natives see 1 (m) other brown eyed native
all others see 2 (m + 1) with brown

meeting m = 2, k natives show up
Since k natives show up, there must be n >= m natives with brown.
the 2 brown eyed only see m - 1 brown eyed, so they conclude they have brown

meeting m = 3, k - 2 natives show up. Since two brown eyed left, the remainder must have blue.


for meeting m, if you see m -1 brown eyes, you have brown eyes and don't show up the next meting.
if you see >= m brown eyes, you cannot conclude you have brown.

Answer 4

We need more information. Is it possible for the natives to give off "tells" that suggest to others which ones have blue eyes? If not, then as long as the devil lets n>1 they will not figure it out.

EDIT: I misread the question. I thought the devil was coming down each night. Now that I've reread it, I can answer it properly. The correct answer is n+1.

Here's why: suppose n=1. Then everybody comes to the first gathering, and the person with brown eyes doesn't come to the second.
Suppose n=2. Everybody still comes to the first meeting. Everybody also comes to the second meeting. When each of the newly brown-eyed people see the other at the second meeting, they will realize that they have brown eyes as well and not come to the third meeting.

Whatever value n is, everybody with brown eyes will see n-1 people with brown eyes. They will then expect that there would be less people at the next meeting if their eyes were blue. When there aren't less people at the next meeting, they will know their eyes are brown so they won't attend meeting n+1.

Will you take us seriously now?

EDIT 2: I had a feeling you still wouldn't take atheists seriously. I solved your homework problem for you, yet I'm still in the list of people who you claim you don't need to listen to. Funny how that works.

EDIT 3: I recant EDIT 2. And you're right about Salient's proof being a better explanation.

Answer 5

Alright... to sum up ALL the responses of value so far.... complete with a fair amount of other logic and personal experience based on the language definitions thus put forth.

#1. If the natives cannot communicate, then "gatherings" can only be coincidental and the probability of all the natives on the island being in a position to judge all the others at any given time is negligable.

#2. Communication is part in expression and part in interpretation. Communication would develop itself over time if the natives were able to encounter each other and able to interact in any way, shape or form. As such... if there is by definition NO form of possible communication between them, then it isn't possible for them to interact or even acknowledge each other's existence in any way, shape or form. As such it is physically impossible for them to gather.

#3. You claimed the natives are on an isolated island, and yet there are no reflective surfaces. By classic definition, islands are surrounded by water... and water is reflective. Likewise... water is conventionally required for survival. If there is no water on the island, how do the natives survive and what is the island surrounded by?

#4. Without any form of communication or interaction with others... where did the natives come from? Did they have no parents?

#5. Without any form of communication or interaction with others..... how do the natives even have a concept of "blue eyes" or "brown eyes"? All they have ever known is blue eyes... and that change is possible likely never occurred to them. As such... knowing that it had happened wouldn't let them know WHAT it was they were looking for, nor any relevance to it.

#6. As previously noted.... assuming everything you've said so far .... the islanders would have no definite means of determining that their eyes had turned brown under any circumstances.
Even the circumstance of one native seeing everyone else's eyes as "blue" would be irrelevant if they had never seen their own reflection and didn't technically know what "blue eyes" were. For all they knew, what we see as "blue" they might think was brown and vice versa... thereby meaning that they believed themselves the only individual justified in actually still coming to the meetings.

#7. Assuming that the natives are otherwise ordinary people..... after an indefinite period of time, they would all stop coming to the meetings due to simple uncertainty and the speculation of whether their eyes had changed or not... which would at some point invariably lead them to each conclude that their eyes were brown.... thereby making the decision not to return to the meetings.

#8. For that matter.... assuming NO communication or interaction between the natives.... each native most likely assumes they are the only inhabitant of the island... and thus if greater than one of the inhabitants of the island has brown eyes, then they must have brown eyes and therefore they will never show up to the meeting in the first place.




*Coughs*

Nice try..... but I still fail to see what point you're trying to make with this nonsensical mess.

Answer 6

This is an old one so I won't take credit for solving it or taking up time and space with the solution. It's probably on teh interweb somewhere. What's interesting is that this is such a familiar logic puzzle and searcher has cribbed it in pretty much a standard form - there's no indication that for all his gloating and criticism, there's the slightest chance that he could do this puzzle without help. It's actually not that straightforward - but it's basic inductive reasoning. It assumes the natives are all good at logic, and who they can see and what they do if they don't see someone with blue eyes. Eventually there are no more meetings.

Answer 7

How would the natives conclude they have brown eyes if "the natives cannot communicate with each other in any way" and they have no mirrors or reflective surfaces?

Even a facial expression would be a form of communication.

Unless, there is only one native who can deduce that he is the only one who has brown eyes because none of the others do.

Edit: Vishal's logic only works with up to three brown eyed people. One could conclude he is the only one after one night. Two, after two nights. Three, after all three realize the other two must be seeing there are at least two others, so why aren't they leaving? After three, no such deduction can be reached.

Your welcome. I understand now why Vishal's is correct assuming all of the natives have the same deductive reasoning. And thanks for demonstrating how you can falsly interpret information based on a small amount of information.

Answer 8

Obviously it is some kind of math formula but its easy enough that if the devil changed n > 0 but communication being n < 0 in all ways even expressions or interaction n < 0

When n < 0 > n
BANG the island blows up.

Answer 9

your hypothetical situation makes very little sense.

id need to know why they cannot see reflections in water, and how a society of natives thrive without any form of communication whatsoever. furthermore, how would they all hold the same idea of the devil changing eye colors of some, if they cannot communicate. they would each hold seperate beliefs on why this event occurred.

since they do not communicate there is no social structure.

thus you wouldnt have any reason for some to avoid others any more than normal. they cannot point at eachother as this is a form of communication.

furthermore, its very difficult to see eachothers eye colors at night. so i dont see why it would be a problem for any of them to come to their night meetings.

why do they come together in a circle at night if they cannot communicate? what purpose would that serve?

i dont see why any of them wouldnt come. regardless of eye color.

furthermore how did a group of natives on an isolated island learn of the devil, to do so you would have to be introduced to the bible. and know how to read or have someone tell you about it. since they cannot communicate they cannot know of this devil.

this isnt including the errors that exist only in this hypothetical story. that is, eye colors dont change, and the devil would actually have to exist. furthermore the devil would have to have the ability to change persons genetic features, which according to the bible is wrong too.

logically, your story makes no sense, and even if it has an answer that your giggling about because your teacher told you this. the story is absurd.

you want logic to answer your question, then logically your story cannot exist as a truth. and if it doesnt exist as a truth, then what purpose does having an answer to a non-truth serve?
....

edit: i already solved it dude.

without communication they have no reason to not show at the meeting. there is no society since there is no communication, therefore they cannot be social outcasts.

unless your withholding information from me?

im not atheist first off. second off, you didnt upset me, but this questions is stupid and pointless.

since you dont understand - God is a contradiction, a contradiction cannot exist.

A = B = C
A != C

is a false statement, and cannot exist as a truth.

God hates sin.
God cannot sin since he is pure.

murder is a sin.
flooding the earth is murder of millions.
god murdered.

thus god sinned and god didnt sin at the same time.
one of many contradictions.

Answer 10

Okay, I'm going to try this WITHOUT looking at any of the other answers first.

If there is one native with brown eyes, he will see that everyone else has blue eyes, therefore he must have brown eyes, so he drops out after meeting 1.

If there are two natives with brown eyes, each knows that they could possibly have blue eyes on the first meeting. However, the presence of the other brown-eyed guy at the SECOND meeting proves that he also sees someone with brown eyes, so the two brown-eyed natives drop out after meeting 2.

If there are three natives with brown eyes, each knows that they could possibly have blue eyes on the first meeting because they see other people with brown eyes. On the second meeting they still see others with brown eyes. On the THIRD meeting, however, they know that anyone who saw only one brown-eyed person should have dropped out by now, so they realize that the other two people with brown eyes must be seeing ANOTHER person with brown eyes, which must be them as they don't see any other brown-eyed people, so all three of them drop out after the third meeting.

And so on. Someone who sees only three brown-eyed people in on the fourth meeting will drop out. Someone who sees only four brown-eyed people in on the fifth meeting will drop out. The brown-eyed people all keep on coming to the meetings until as many meetings have passed as there are brown-eyed people, upon which they all drop out. In terms of n as the number of brown-eyed people, the nth meeting will be the last for all the brown-eyed people, and the n+1th meeting will be the first meeting with only blue-eyed people.

Do I get a cookie now?