Some of us might be familiar with the blue eyes riddle. We also might be familiar with the "solution". If you aren't - stop reading immediately and educate yourself, man.

What few people know is the answer to this 'riddle about a riddle':

- What information does the Guru give us?

(so you could perhaps say, that few people really understand the solution) 

This is non-obvious. If there's, say, 50 blue eyed people on the island, the Guru clearly tells us nothing new about the situation (we can see "someone" has blue eyes, duh!). However, I can say with confidence that without the Guru telling us "someone has blue eyes", nobody would've been able to leave the island.

Why is this?

Take some time to think, maybe discuss it in the comments, if you like. You can spoil yourself the the answer here.
"I can see someone with blue eyes."

...who I may have mentioned yesterday?
Well, the Guru only speaks once, ever.
What does the riddle mean by "on what night"? I don't get how you figure that out. They leave whenever they figure out what color their eye is, what does it matter which night they leave?
The Guru is allowed to speak every day, saying "I can see someone with blue eyes."

It doesn't say he speaks every day, he's just allowed to.
It also doesn't say he isn't repeating who he's talking about.

Other than that, I do understand the solution.
at EmpirezTeam

The reason why "on what night" is important is because none of the villagers with the same eye color(in this case, the ones with blue eyes, because that is what the guru specified.) can leave at different times.

All the people with blue eyes have to leave at the same time because that is when the conclusion (for people with blue eyes)is reached.

Really, by adding the addition of "on what night" shows that the reader has additional understanding of the concept versus just guessing the semi-obvious answer of "100 blue eyed people"

at Kaiochao

Even though the guru only speaks once, its not important who it was directed at because for blue eyed people, they only can see 99 blue eyed people, if after 99 days, they still see that blue eyed people are still on the island, they can conclude they are also blued eyed (hence, everyone who sees 99 blue eyes on the 99th day, also has blue eyes, they all reach this conclusion at the same time).

(this makes sense because brown eyed people see 100 blue eyed people, and on the 99th day, they still see 100 people with blue eyes, thus they can't yet reach the conclusion they have blue eyes (and when the 100 blue eyed people leave, they can conclude they do not have blue eyes, because the other blue eyed people wouldn't have left had they seen that you had blue eyes.))
What I'm saying is that with 100 blue-eyed people, the guru can say "I can see someone with blue eyes" and be talking about the same person every single day. No one knows whether the guru is or isn't, because the riddle doesn't specify.
Except the guru can only ever speak once in his or her lifetime.
At Kaiochoa

Quoted directly from the Riddle:

"The Guru is allowed to speak once (let's say at noon), on one day in all their endless years on the island."

This states that the guru can only speak once.

Even if the guru could say that everyday, it doesn't matter whom is it directed at, because just having the fact "someone has blue eyes" provides a constant, which will inevitably lead to the conclusion that the blue eye people will be able to leave.

(which could lead to a trust problem, because what would she/he say when all the blue eyed people leave?)
I definitely read the riddle wrong, like three times.

Solution is that since no one leaves, everyone looks at each other's not-leaving and leaves?
At kaiochoa

Not quite, what this suggesting is that because everyone has the exact same mindset, the only difference between people is their eye color.

The key difference plays in that blue eyed people only see 99 blue eyed people. brown eyed people see 100 blue eyed people

the solution essentially is that once the number of blue eyed people you see is equal to the number of days that have passed, means that you have that color eyes and can leave at the next possible chance.
The description threw me off. I was thinking from the perspective of one of the people where it said that one could have red eyes, and there could be 101 blue, 99 brown. Gah, why do riddles have to be so hard?
Well, guys, you seem to be discussing the riddle but none of you have tried thinking why what the Guru says matters. Why would this riddle be unsolvable without the Guru? Does she tell you anything new? I mean, you can already see there's 99 blue-eyed people before she opens her mouth.
At Toadfish/original post

What the guru says is important because it acts sort of like the "constant" of the riddle if you want to think of it in mathematical teams.

Sure, you can also visual see that there is 99 blue eyed persons (assuming you are also a blue eyed person) But you need that constant to have a place to start from, otherwise, your just stuck in a large cycle.

The reason why you can't use the day that your inducted (or born) is because you can count 200 people, and you can count 99 blue eyed people, and you can count 100 brown eyed people, but that doesn't help you because it doesn't present a starting point everyone can reference too.

The day the guru says that someone has blue eyes, the process can begin because, sure you already knew that, but now you (and more importantly, everyone around you (who all have the same mentality that you have)) can begin the process.

This was slightly touched on in a few comments below when I was saying that it didn't matter who the guru was directing the statement at, or how many times the guru said it. Just the fact that the guru said the statement at all is what matters.
Toad: I have a question on the rules of the original. If I can get this fixed up, I'll be able to think clearly.

There are 201 people, 100 of each color and 1 Guru. However, the people do not know the rule that it's 100 each, correct? They may believe that there could be 101 of 1 blue and 99 of brown, and vice versa?

Edit: Excellent. I thought about it more and realized that question has nothing to do with it. I got it right. ;)
They can basically see everyone's eye colour but themselves, and anyone's eye can be of any colour. The solution works if there's 56 brown, 2 purple and 42 blue too, for example.

Higoten: very close, but it goes a bit deeper than that! Notice that from your point of view, everyone knows there is at least one person with blue eyes. So we definitely have a 'common grounds' in that regard. The real 'common ground' the guru gives you is a bit more subtle than that (you can read my explanation in the main post).
I solved it and checked the answer and was right, but a useful hint I thought of to derive the answer was that the Guru was an external being. Their eye color not mattering was incredibly useful.
Whatever floats your boat! But I mean, the Guru is one of the residents, so if she had blue eyes she'd have left too. At least that's how I interpreted the riddle. And in fact, the riddle could still be solved had she begun with 'I have blue eyes.'

I think the single most useful hint to really understanding the solution is thinking of the riddle in terms of a chain of hypotheticals: 'let's say I have brown eyes, what would the person next to me do on the Nth day?'
Indeed. Telling them they had blue eyes was extremely useful for one reasons:

1. They wouldn't have a starting point to work from.

It's purely psychological. Imagine if the question hadn't been asked. They couldn't predict if anyone was going through the same thought process as them. For all they know, one could be considering "someone has brown eyes".

Also, it is relevant to note why only 100 blue leave on the 100th night. This is because there could be more than brown eyes. Red, Yellow, Green, Pink - Anything. Hence, when the brown are left, they are unable to figure out which they are.

It's good to pretend what if it was 100 blue and 1 brown. The same would occur, if he said "I see someone has blue eyes", they'd go through the same process. The number of blue doesn't matter, nor the plethora of colors. As long as there is just one blue this can be solved.

They have to wait, because each day asks a different question in there head. The day the Guru speaks, each waits to see if anyone leaves. Obviously nobody will, but they still wait. When they don't leave it states "No [TotalAmount-1] non-blue people exist". Essentially, there isn't 199 brown. The next, there isn't 198 brown. This continues until "No 101 non-blue people exist". Once this statement occurs, he says "Well there is 100, so that means I must be blue!". And the blue person leaves.

Admittedly, I was tired and decided to look up the solution to the initial riddle. I'll show the work I did till then. It was mainly lots of "ifs".

If blue and consider yourself blue, then another blue will see the following: 99 blue, 100 brown

If blue and consider yourself brown, then another blue will see the following: 98 blue, 101 brown

If blue and considering yourself [x](Other color), then another blue will see the following: 98 blue, 100 brown, 1 [x].

etc etc...

By finding patterns in the combinations, you see ones that infinitely repeat, ones that converge onto someone thinking "I'm blue!", and impossible ones with 199 colors. Haha

From there, I figured that the pattern that converges is similar to a recursive program. It'll kind of feed back on itself.

Sadly, my own idea seemed so insanely crazy I figured to just look at the answer. Sorta wish I just took the time.

More importantly! I hope that I understood the answer adequately! Funniest hour of sleep deprived thinking of my life :D
You'll be glad (sad) to know there is nothing psychological in the works! There is actual, concrete information embedded in the Guru's saying "I see someone with blue eyes", despite you also seeing someone with blue eyes, and you knowing everyone else can see someone with blue eyes, the Guru included. Even if they all knew the solution to the riddle (with an arbitrary amount of blues and browns), they still wouldn't have been able to leave the island without the Guru speaking.
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