Logical And Wrong

So, continuing my last post about monks with eyes, (though I should start calling them islanders now), if everybody assumes that their eye colour is one they have seen before, then they actually can figure out stuff fairly quickly. Its interesting to see what happens if that information is wrong though.

For example, assume there is 1 blue-eyed person and 3 brown-eyed people. The blue-eyed one will have to assume his eyes are brown and leave that night, and the brown eyed people will figure out it the next night and leave. Of course, the blue-eyed person will leave because he thought he figured out his eyes were brown and he is just wrong, but thats what happens when perfectly logical people are told the wrong information.

Continuing upward, if there is 3 blue and 2 brown, the 2 brown leave on the second night and the blue leave on the third night, its easy to see this because a brown eyed person will assume his eyes are either brown or blue.

Next, assuming there is 1 green, 2 blue, and 2 brown, the green-eyed person will assume his eyes are either blue or brown. If his eyes are blue, the brown eyed people will leave on the second night and vice-versa for brown. When the second night comes and nobody leaves, the green eyed person must realize a contradiction.

At this point, you can't really guess what happens next, when perfectly logical people see a contradiction they might come to second guess everything, so we really have no choice to declare there is a contradiction and end there. As a side note, the blue and brown eyed people all know that somebody has seen a contradiction, but they cannot figure out who knows it.

After seeing this, I came to a conjecture:
Either somebody will realize there is a contradiction and nobody leaves, or everybody will eventually leave the island.

Its not hard to justify this, as if anybody leaves, then everybody else gains a big piece of information, but if a contradiction is seen and somebody doesn't leave in time, then nobody can gain that information (unless, I suppose, if the person who found the contradiction is allowed to announce it).

Last neat thing I want to say about this puzzle is what happens if you give the assumption that no person has a unique eye colour. This is slightly different than the assumption that each person has an eye colour they have seen before (in spite of the fact that both are true under the same circumstances, which still sort of confuses me a tiny bit, to be honest).

In this case, with 2 brown and 2 green, they all leave on the first night, seeing that they all can see an eye colour that only one other person has. And with 1 blue, 3 brown, and 3 green, the brown and green will assume their eyes are blue the first night, and will all leave, and the blue guy will simply be left asking where everybody went. It appears my conjecture does not hold here anymore.

Alright, I'm done with that class of monk problems (for now). I'll start something new next time.


James said...

You're logical and wrong.

Anonymous said...

doesnt really add much, but the "have to have seen that color" thing and the "unique eyes" thing will turn out the same. in the first example you gave you didnt consider that each of the blue and brown eyed people will assume their eyes are green, since there is clearly one green eyed person, and that person must have seen a different green eyed person, ie "you". so they'll all leave and then there wont be bridge


hairguy said...

also, the fact that your conjecture is wrong is really just because the problem sucks. if there is only one green eyed person, and nobody knows the set of possible colors, then that guy will never think "oh my eyes are green" no matter what the circumstances are. i think it might work out if you allow process of elimination to cause leaving. or rather, if you had to leave when you determined how many people have a particular eye color (which will turn out to be the same problam when the colors actually arent unique)

kstevens said...

Well, the way I analyzed it, the two assumptions came out different because the first case allows the monks to assume the the other people are wrong in their assumption, whereas the second did not.

For the other thing though, I will admit the problem does sort of suck, however, even if their is only one green eyed person, he might leave if he says "oh, my eyes are blue". He will be wrong of course, but thats the whole point of the title.