# Variant to blue eyes puzzle - selfish logicians

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?

• Not sure I understand the nature of their 'selfish' behaviour if they will help others leave.
– Bob
Commented May 15, 2015 at 17:31
• Do you know the answer or is it a genuine question? Commented May 15, 2015 at 17:45
• @BmyGuest Regardless, it is still a question :) Commented May 15, 2015 at 17:47
• @MarkN Fair enough, I was referring to site terms which refer to "Question" as opposed to "Challenge" ;c) Commented May 15, 2015 at 17:49
• 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. Commented May 15, 2015 at 20:45

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.

• 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. Commented May 15, 2015 at 21:27
• also, "no one knows who has given the statement" Commented May 15, 2015 at 21:29
• @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. Commented May 16, 2015 at 1:55
• Since everybody knows the rules who will trust the statement of the logician? Commented Feb 21, 2016 at 21:42
• Since the brown-eyed people aim to maximize the number of people getting out, which includes the Guru, if the person picked on the first day is brown-eyed (so brown-eyed people are still on the island on the second day), they will make a statement "Everyone will have green eyes tomorrow" (or even a false statement "Everyone has green eyes") to hint the Guru that she has green eyes. If the first person picked is blue-eyed, then if I interpret the selfishness rule correctly, they won't make any (useful) statement on the second day, leaving the Guru stuck. Commented Jul 22, 2021 at 7:11

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.