The blue-eyed islanders puzzle
Saturday, February 16th, 2008Yesterday:
[Quote:]
There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).
Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).
One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.
One evening, he addresses the entire tribe to thank them for their hospitality.
However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.
What effect, if anything, does this faux pas have on the tribe?
So……
It helps to make the numbers smaller.
There are ten people on the island, one blue eyed, the rest brown. The guest says it’s nice to see another blue-eyed person. Blue-eye kills himself the next day because he can see nine brownies, thus the guest must be talking about him.
Next up. Ten people, two blueys, eight brownies. The guest stirs shit up. bluey1 thinks the guest could be talking about bluey2, so does not kill himself. bluey2 thinks the same thing, so does not kill himself. When neither kills himself the next day, they both know that the other bluey must be seeing another bluey, or they would have offed themselves! Which means that since they both see eight brownies, they must both be the other bluey. Both kill themselves on day+2.
Etc.
Now back to the original numbers:
Assume (and given the premises we’re dished, this isn’t a far cry) that these villagers all know one another. So you have two cases:
1. Villager is brown-eyed. He looks at (or devoutly recalls) the eyes of his 999 friends and neighbors and knows that 100 of them have blue eyes and 899 have brown. His own eyes, he does not know; if he is brown-eyed, there are 100 blue-eyes. If he is blue-eyed, there are 101 blue-eyes.
2. Villager is blue-eyed. He knows that there are 900 brown-eyes and 99 blue-eyes, and him. If he is brown-eyed, there are 99 blue-eyes. If he is blue-eyed, there are 100 blue-eyes.
So browns know there are between 100 and 101 blue-eyeses; blues know there are between 99 and 100.
Following the stated induction solution, any given blue will know that there will be a mass suicide on the 99th day if he’s not a blue-eye; and so on the 99th day he doesn’t not commit suicide, as he still doesn’t know his eye-color. Thereafter, when there is no suicide (because every blue-eye has to this point reasoned thus), he will know there are not 99 but 100 blue-eyes, and that he must therefore be a blue-eye. So he kills himself on day 100.
Each brown-eyes has been, by the same reasoning, sweating the hell out of day 100 and hoping (if hating himself for it) that all the blue-eyes he knows kill themselves that day, because he has reasoned that if day 100 isn’t a mass suicide, there must be a 101st blue-eye. And that’d be him, and he’d be killing himself on day 101.
And the day after the mass suicide, each brown-eye realizes the color of his own eyes is brown, and promptly kills himself.
Sounds logical, right?
Unfortunately, it is also wrong. Let’s go back to the smaller numbers, because I fooled you all with the “Etc” bit.
So — another example ten people, four blueys, six brownies. The guest stirs shit up.
Day one rolls around and everyone on the island can see at least see at LEAST three blueys. Let’s call them A, B, and C. Any given observer will reach the following conclusions:
A can see both B and C so he has no reason to suspect that he’s a bluey
B can see both A and C so he has no reason to suspect that he’s a bluey
C can see both A and B so he has no reason to suspect that he’s a bluey
Because of this mutual deadlock nobody is expected to commit suicide and nobody does.
Day two rolls around and the same deadlock exists.
Day three ….
Day four …
Day five …
Etc.
In short, when you have at least four blueys, you immediately reach the point where no one is committing suicide and everybody knows why that is. And it doesn’t matter how many days pass.
Or, to put it another way: the tourist didn’t give them any new information at all, so why should things change?
