On an island, there are 100 people who have blue eyes and 100 people who have brown eyes. No-one on the island knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island that night. On the island, each person knows every other person's eye color, there are no reflective surfaces, and there is no discussion of eye color. Also it is public knowledge that the islanders are perfect logicians - if a conclusion can be deduced then they will immediately do so.
One day, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement:
"I can see someone with blue eyes."
Who leaves the island and on what night do they leave?
If you haven't seen this puzzle before, I encourage you to think about it before continuing.
...
All done?
Is your answer that no-one will leave the island? If so then I'm sorry to say that you are incorrect! Despite the fact that all the islanders can see many blue-eyed people, the outsider's statement does convey new information to the islanders and that information enables them to determine the color of their own eyes.
I will give the actual answer and explanation in a few days time. In the meantime, here is a suggestion to help you find the correct solution.
First try to solve a much simpler version of the puzzle. The simplest version is the scenario where there is one blue-eyed person and one brown-eyed person. Consider what the islanders can deduce before the outsider arrives. Then consider what they can deduce after the outsider makes his statement.
Now solve the next simplest version where there are two blue-eyed people and two brown-eyed people. Again consider what the islanders can deduce both before and after the outsider makes his statement. This will help you notice what the islanders learned from the outsider's statement and how it is necessary for solving the subsequent versions with three blue-eyed people and so on.
[Added Sep 22: The Blue Eyes Puzzle solution]
There was an exercise run with a group along the lines of: everyone pick a number between 1 and 100, with the goal being that your number is 2/3 of the average of all numbers picked" (wiki: Guess 2/3 of he average).
ReplyDeleteExperimentally, the results tend to be around 20 which is three iterations of reasoning.
Likewise, unless your island has very dedicated rationalists, I expect that the arrival of the stranger would cease to have an impact once the population was more than, say, 8 or 10 people.
Not that it isn't an interesting puzzle, but at a population of 200, it just seems a little ... fantastic.
Yes, the puzzle is premised on the islanders understanding the full logic of the situation. Otherwise, as you expect, no-one would leave! I've updated the puzzle description to reflect that.
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