26th March 2007, 12:05 AM
Great Rumbler Wrote:Nope, I just don't get it.
Well the key thing to remember is that all the islanders are fully aware that all the OTHER islanders are perfect logicians and that the other islanders know everyone else's eye color.
So here's the deal. If you have two people with blue eyes and everyone else has green eyes, each of the people with blue eyes know the following: "I know of only one other person with blue eyes". "The other person knows exactly what my color eyes are, and would instantly conclude they have blue eyes if they knew that everyone else on the island did NOT have blue eyes." "If they don't leave that night, it can only mean they did NOT have enough information to conclude they had blue eyes." "The only way that could possibly be the case is if someone else had blue eyes." "Since I know everyone besides that person has blue eyes, I must conclude that the reason they didn't leave is because I have blue eyes." "I have blue eyes." Keep in mind that BOTH are going through this train of thought.
Now, let's extend that to three people. The difference here is that one night isn't enough, but they all know the above sequence would be true if only those other two guys had blue eyes, and those other two would both have to leave the second night if that was the case, and the only reason they would not have is if the person thinking about it was still confusing them because they too had blue eyes.
The rest follows all the way to 100.
"On two occasions, I have been asked [by members of Parliament], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able to rightly apprehend the kind of confusion of ideas that could provoke such a question." ~ Charles Babbage (1791-1871)