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# Behind Blue Eyes...

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Alternate Title: Brown-eyed girl.

Sorry to disappoint you guys, (or in a few cases a...ppoint? i don't if that's how that word works), but this entry is about math, not music.

Or, more accurately, it's about logic. Which is just math, but with more numbers.

So, I recently rediscovered the blue eyes puzzle, in all of its mystical wonder.

The problem goes a little something like this: (quote stolen from Randall Munroe. Please don't sue me.)

A group of people with assorted eye colors live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.

The Guru is allowed to speak once (let's say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:

"I can see someone who has blue eyes."

Who leaves the island, and on what night?

The answer to this problem is a lot harder to find, and a lot weirder than you would initially think. Think about it for a while, if you solve it before you crack and look at the spoiler below then you're one of the best logicians in the world. Good job.

All 100 blue eyed islanders will leave on the 100th night after the Guru speaks.

I can hear you through your computer screen.

"What!?" you say, speaking aloud instead of typing like a normal person. "This Eeko guy doesn't make any sense? He's terrible at logic!"

Well, you're right. I'm not making any sense, and I'm pretty bad at logic. But it's still right.

However, this is not actually what this entry is about.

"What new information did the Guru give?"

This question is a little easier to find, but a little harder to prove definitively.

When you think about it, the Guru doesn't seem to give any new information. I mean, everyone one on the island can see blue eyed people, so why does having the Guru say it change anything?

So here is my challenge to you guys and girls.

It's much easier to solve it than the first question. (I was able to after much thought and a little hair pulling.)

Or maybe the day after, to give y'all more time.

I thought it was fun to work through, so you should give it a shot.

Peace.

P.S. I think this is the most words I've had in a blog post before. Woohoo.

... If these people are so good at solving logical problems, why don't they just all tell people their eye color so everyone can go home? How do they solve logic problems if they can't communicate? How do they live if they can't communicate? Seems rather silly to me.

And that's about as far as I got with trying to figure this out.

Ow. I have a headache. Even whipping out the logical structuring techniques didn't work for this one.

Troll solution:

Obviously, you are able to confirm whether the eye color which you assume to be yours is correct or not: If it is correct, you can go on board, if not, you must stay and conclude your answer was wrong. Knowing that the ferry will return every evening, you can go through all eye-color-possibilities until you hit with the right one.

Other troll solution:

Hearing the guru speak, the inhabitants realize that communication is indeed possible and proceed to tell each other their eye colours.

Here is another solution which actually makes sense:

It is said that if a conclusion can be logically deduced, they will do it instantly. This means that prior to the guru speaking, there was no way of logically decucing the own eye colour, otherwise they would have done it instantly. However, the guru's statement carries no new information as far as I can see (after all, everybody knows there are people with blue eyes as they have seen them themselves), so, seeing as there was no way to deduce the eye colour before and there is no new information, there is still no way to deduce it and the inhabitants all stay on the island.

-Gata

I got impatient and googled the explanation. The solution does make sense after thinking about it for a while.

It is said that if a conclusion can be logically deduced, they will do it instantly. This means that prior to the guru speaking, there was no way of logically decucing the own eye colour, otherwise they would have done it instantly. However, the guru's statement carries no new information as far as I can see (after all, everybody knows there are people with blue eyes as they have seen them themselves), so, seeing as there was no way to deduce the eye colour before and there is no new information, there is still no way to deduce it and the inhabitants all stay on the island.

-Gata

I'm afraid not. The guru does in fact impart new information to the villagers.

Just read the solution. It makes sense and it's actually not too hard to understand imo, but maybe that's because I study math. Still very tough, I couldn't come up with the solution on my own.

-Gata

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