4
$\begingroup$

This is a variation of the blue eyes puzzle.

Suppose the logicians want to get off the island. They are selfish and think only of themselves. They may help others as well, but solely for their own benefit. However, if a logician realises that he has brown eyes, he will try to ensure that the maximum number of people get off the island.

On the first day, the Guru makes her statement. On every subsequent day, 1 logician is secretly picked at random, and is asked to make a statement. The statement comes via an intercom, so no one knows who has given the statement or if it is true. The statement can not refer to individuals (by name, place, etc.) but must refer only to the following three sets - blue-eyed, brown-eyed, all. It must be logical and can use maths.

Will anyone ever make a useful statement? Will anyone ever leave before the 100th day?

$\endgroup$
  • 1
    $\begingroup$ Not sure I understand the nature of their 'selfish' behaviour if they will help others leave. $\endgroup$ – Bob May 15 '15 at 17:31
  • $\begingroup$ Do you know the answer or is it a genuine question? $\endgroup$ – BmyGuest May 15 '15 at 17:45
  • $\begingroup$ @BmyGuest Regardless, it is still a question :) $\endgroup$ – Mark N May 15 '15 at 17:47
  • $\begingroup$ @MarkN Fair enough, I was referring to site terms which refer to "Question" as opposed to "Challenge" ;c) $\endgroup$ – BmyGuest May 15 '15 at 17:49
  • 3
    $\begingroup$ I suggest adding a full description of the setup, to make this puzzle self-contained. While a link to a related puzzle is useful, it is best to include all necessary information here. $\endgroup$ – Julian Rosen May 15 '15 at 20:45
5
$\begingroup$

According to the puzzle definition, they can only leave if they know their own eye color, and they all want to leave as soon as possible.

So the first logician honestly states "I see X people with blue eyes."

If he has blue eyes, he says "I see 99 people with blue eyes." All the brown-eyed people see 100 people with blue eyes. Each of them knows that if he had blue eyes, everyone else would see at least 100 people with blue eyes, so the first speaker would not have said 99. They all conclude that they have brown eyes and leave that night.

If he has brown eyes, he says "I see 100 people with blue eyes." Each of the blue-eyed people see 99 other blue-eyed people, and know that in order for the first speaker to tell the truth, he must also have blue eyes. All the blue-eyed people leave that night.

The next day, everyone, including the original speaker, knows that everyone remaining must have the other eye color, so they all leave the following night.

This plan allows the first speaker to leave the night of the day after he spoke. There is no way the speaker could learn his own eye color faster; speaking the truth is clearly the optimal path. Since speaking the truth is the optimal path, everyone else will trust him.

Thus, this is what he would do, and everyone of the other eye color would leave that night, and everyone else would leave the next night.

Except the guru. No one would benefit by telling her what her eye color is, so she remains stuck, alone.

$\endgroup$
  • $\begingroup$ this would be easier if the "he"s were less wandery. Sometime "he" means the logician who said something, and sometimes it means the person who is pondered what was said. $\endgroup$ – Kate Gregory May 15 '15 at 21:27
  • $\begingroup$ also, "no one knows who has given the statement" $\endgroup$ – Kate Gregory May 15 '15 at 21:29
  • $\begingroup$ @Kate 1. The answer is clear enough to me atleast. You could always edit it. 2. No one knows who gave the statement, but the answer tells what happens in both cases, and the logic that follows. $\endgroup$ – ghosts_in_the_code May 16 '15 at 1:55
  • $\begingroup$ Since everybody knows the rules who will trust the statement of the logician? $\endgroup$ – Marco Disce Feb 21 '16 at 21:42
0
$\begingroup$

I don't think it would make a difference (not 100% sure):

Since no one knows their own eye colour, they will all tell false statements as their defense (useless) and the only way they people with blue eyes realize they [all] have blue eyes is on the last day (similar to brown eyes when they are left behind). So even if the brown eyes try to help, any (all) of the perfect logical blue eyes should have been gone by then.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.