The following is from TEDEd:
This “green-eyed logic puzzle” kind of reminds me of a logic joke I’ve seen.
Four logicians walk in a bar. The host asks the question, “Do you all want a drink?” The first logician says, “I don’t know.” The second logician says, “I don’t know.” The third logician says, “I don’t know.” And the fourth logician says, “Yes!”
The fourth logician can respond with a “Yes” because if logician one, two, or three didn’t want a drink, their answer would have been a “No.” And if the fourth logician didn’t want a drink, he could have said “No.”
The “green-eyed logic puzzle,” to be sure, is more complicated than the bar joke.
Yet when you consider two prisoners, the puzzle is not too hard. It just seems hard when you have 100 people. So, when it comes to these problems, it is always a good idea to attempt to simplify it and then try to solve it. And if you can do that, you can extrapolate and tackle the seemingly more difficult problem with its greater numbers involved.
In the easy case, since it is known that “at least one person has green eyes,” person A and person B can figure out things and leave on the second night. That’s because person A knows that if person B left the first night, that would only be because B saw that A is non-green. That didn’t happen! Ergo, person A realizes that he must have green eyes.
The same logic applies to person B. Ergo, person B realizes that he must have green eyes. As the number of people increase, the number of days increases proportionally as it takes as many days to observe as there are people so as to watch their actions, and those actions reveal needed information. With three people, it takes three nights. With n people, it takes n nights. We can see this modeled after induction by generalizing from two people to n people.
The video references David Lewis (who is famous in philosophy for his ideas about “possible worlds” in logic) and suggests that the “common knowledge” being shared to everyone at once makes a difference. It makes a difference because now everyone is keeping track of everyone else based on the “common knowledge.” In other words, the video suggests, for everyone to keep track of everyone during these days with a success rate of 100 is for the simultaneity of watching at that point when the “common knowledge” was broadcasted.