Quoting Terry Tao, albeit with small modifications,
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).
For the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.
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?
Although, it might not initially be obvious this is a problem about thinking about thinking about thinking about thinking — and so on. As before, I will proceed with diagrams. One argument, which is inductive, proceeds in this way: if there were only one blue-eyed villager, he would not know his eyes were blue. Everyone else would know his eyes were blue. Because the stranger pointed out there was an individual with blue eyes, an individual our sole blue-eyed villager had never seen, he would immediately know it was him and kill himself the next day.
In the case of two blue-eyed people. Let us assume, they would tend to hope for the best, holding in their minds the optimistic conjecture, that they do not have blue eyes, unless facts prove this conjecture wrong. Each blue-eyed person knows he has seen one blue-eyed person in his life. He assumes the best and thinks that the blue-eyed person is the only blue-eyed person. In this case, he thinks the situation in the first diagram is true. However, in the first diagram the blue individual knows immediately he is the only blue-eyed person and kills himself the next day. However, because there are two blue-eyed people, each one is waiting for the other to kill himself. That is to say, there are two blue eyed people thinking that there is only one blue-eyed person whom they think thinks he must be the only blue eyed person. When the first day goes by and no one kills themselves, it becomes obvious to everyone that there are actually two blue eyed people.
More complicated still is the case of three blue eyed people. Each blue eyed person hopes for the best, that they don’t have blue eyes. They don’t know because, as we said before, they have never seen themselves. In this case, they see two blue eyed people. Their mental model for reality is that there are two blue-eyed people and further, they can have mental models of how these blue-eyed people think. They find themselves thinking these blue-eyed people must think like the people in the previous diagram. These hypothetical blues don’t know their own color. They assume the best, that they are brown-eyed and that the other blue is the only blue and so on.
It’s intriguing isn’t it? The nested models of thinking. Each of the three constructs a mental model of the two others. In those mental models, each of the two think of the remaining one.
For models of five, six, seven, as many blues as there are, it goes on like this. One blue kills himself in one day. Two blues kill themselves in two days. Three blues kill themselves in three days. All the while, there is thinking about thinking about thinking.
If the browns know that there are only two possible eye colors, on the day after the blues all die, the browns die too.
Such fearful symmetry.