It's a funny thought to imagine ten people waking up one morning and all simultaneously realizing their own eye color upon observing that the status quo has never changed
So except for the case where there's only one blue-eyed person (in which their existence is actual information to them) the stranger just acts as a starting signal for the day counting. Which means that if everyone already sees more than one blue-eyed person, they could just agree together on a day that they start counting… actually this would even work for the 1 case, because simply the brown-eyed ppl initiating this agreement would replace the stranger's statement as information for the only person who didn't already know! I guess this would mean allowing this "special meeting" as an exception to the "we don't talk abt eye color" rule, so the stranger's introduction is more natural, but still!
Stranger gives you the knowledge that other people have the knowledge that all of you knew but didn't know that they did. The stranger is essential as he isn't starting an arbitrary countdown, he's converting converting Nth Order Knowledge to Common Knowledge.
It should be pointed out that while the puzzle lets you know there are 10 people with blue eyes, that is not information the islanders have. They don't know how many people have blue eyes. But since they all see at least one person with blue eyes, it appears the stranger's statement doesn't add any new information. But, in the first walk through where there's only 1 person with blue eyes, the stranger actually does give out information that wasn't known to everyone. The question really becomes why didn't the people with blue eyes leave the island on their own after 10 days of the rule being in place?
Well... Lets see If there is no stranger in this puzzle, what will happen? In the case there is only 1 person with blueeyes, of course he/she will not leave on his/her own. In the case there are 2 persons with blueeyes, they also will not leave on their own .In the case there are 3 persons, they also will not leave on their own.... All because they are not sure about their eyes color and they know everyone else also doesn't know about their eyes color... This puzzle is so weird =]]
They don't know that at least one person has blue eyes. What if there's none? No one should leave If they knew that there was at least one person with blue eyes (that's basically what the stranger told them), they would leave all on say ten
It wasn't made clear that the islanders do not know how many there are with blue-eyes. Couple of the comments I see here were thrown off by that. It's also probably important to include that if and _only if_ an islander learns they are blue-eyed, they leave the island. I know there's an effort to save space and words for simplicity sake, but there's also some key details too important to only include in the voice-over and not the text. Specifically, the islanders being forbidden to talk about eye colors and that they have the opportunity to leave (via ferry) _every_ midnight. I think this kind of defeats the purpose of showing text since it looks like everything you need to pause and think about the question, but will just leave the reader confused if they didn't pay attention to the voice-over.
However, I assume that if they didn't have previsous knowledge of the amount of blue-eyed people, the brown-eyed people would think the same, so they would also leave, right? Correct me if I'm wrong btw.
If there are 10 blue-eyed people and you are one of them you see 9 blue-eyed people. On day ten the 9 havent left which means (using the logic from the video) there are 10 blue-eyed people and you are the tenth. On the other hand, if youre brown-eyed you will see 10 blue-eyed people and they will wait until day 11 to see if they leave. And they will leave
@@sinisteak6790 That's a funny way too prank them, the blue eyed people decide not to leave, and every brown eyed person thinks "Oh they must be waiting for me, i must be the last one with blue eyes!" and leaves
The statement of the puzzle in isolation seems ambiguous to me. There are 10 people with blue eyes, but crucially nobody on the island knows the total number of people with blue eyes - if they did they could simply just count how many people they can see with blue eyes. It would be more accurate to just say "Everyone on the island has either blue or brown eyes", rather than "10 people have blue eyes, and many have brown eyes". When you state the puzzle in this way, it clearly shows how the stranger is definitely not "saying nothing" when they say that they can see at least one person with blue eyes because it explicitly rules out the possibility that there are zero people on the island with blue eyes to begin with. Aside from that it's still an interesting puzzle and a cool inductive solution for any arbitrarily sized island/number of blue-eyed people.
I almost broke my brain trying to fit all this in my head, and it's gonna be terribly phrased, but it's basically many levels of abstract assumptions. Person 1 assumes that they have brown eyes, and there are 9 people with blue eyes. They know that if there are indeed 9 blue-eyed people, Person 2 would have to assume that there are 8 blue-eyed people (of course they know that this isn't the case). Then based on this assumption, to learn their eye colour, Person 3 would have to assume there are 7 blue-eyed people etc etc. Of course everyone knows that there are 9 other people with blue eyes, but this fact won't get them anywhere. They can only learn if there actually are 9 or 10 blue-eyed people by following this wild chain of assumptions, and disproving them one-by-one. Which includes assuming that someone assumes that someone assumes... that there are no people, which can only be disproven by this stranger's statement. It's really hard to wrap your head around all this, but it all comes down to ignoring the fact that obviously there aren't fewer than 9 people with blue eyes, and following this train of thought that actually allows you to find out if there are more than 9. Sorry if this sounds needlessly convoluted.
They aren't allowed to talk about their eye colors. So, everyone knows there's at least 9 blue-eyed people, but as long as no one brings it up, it doesn't matter. So, essentially, they're all pretending they see nothing, but the stranger's words mean they don't have a valid excuse anymore.
I paused just before you stated that everyone having perfect logic is common knowledge, and i was JUST about to comment that assumption is EXTREMELY important.
Wait. In the last example with 10 people with blue eyes, does the stranger say that there are 10 people with blue eyes or that there's at least 1 person with blue eyes?
From my understanding, they would say "at least 1", but I don't think it really matters, whatever is the number of blue eyes people, let's say n, if you have blue eyes you see n-1 blue eyes and if there would have been n-1 blue eyes people, they should all leave the n-1th night, and since it's not the case you know that there is actually n blue eyes and you have blue eyes
Ok, the phrase "after the stranger makes the statement, it becomes common knowledge" is driving me completely insane. I think it's a horribly misleading phrasing, probably with a specific meaning to logicians (but if so, it's not adequately explained, along with "9th order knowledge"). Would it be accurate to say that, as perfect logicians, the stranger's statement "starts the clock", so to speak, and everyone knows that once the day counter PASSES the number of blue-eyed people that each blue-eyed person sees (which happens one day sooner than for those with brown eyes), then each blue-eyed person self-identifies? If so, then why not just say they were all dumped on the island at the same time, with the same rules. The ONE AND ONLY time the stranger matters is in the case of only a single blue-eyed person! In every other case, everyone starts out knowing what they stranger said, so claiming that starts the clock seems completely unnecessary. All you have to do is identify a day 1, and this way seems to be intentionally misleading, in a way that is simple to avoid. Still, have to give credit for explaining the puzzle well enough that I finally understood it. I know the terminology and set up wasn't chosen by this channel.
@@junj1023 Yeah, I'm aware. In the ten person case, everyone already knows there is a person with blue eyes, though. Everyone knows there are at least 9 blue eyed people.
Of course, this knowledge can be expanded on indefinitely. Even if there was a massive space prison with thousands of blue eyed people and the same rules, the single statement from the stranger can eventually release every single blue eyed person from their prison… even though it will take a few years to get to that point.
This makes sense for the examples with 1 or 2 people, but for 3+ people, wouldn't they just go, "Huh, they did not leave. I guess they assumed he was talking about the other guy."
The idea for 3 people is that because everyone is perfectly logical, the two blue eyed people would have left on night two if they looked around and saw only one other blue eyed person. But, since they didn't leave, you learned that there is at least one more blue eyed person on the island, totaling to 3. If you don't see the third person with blue eyes, it must then be you that has blue eyes, so you need to leave too. Therefore, on night 3, all the blue eyed people leave. The same logic extends past three, just taking an extra night per extra person with blue eyes. For example, let's run through the logic if there are 6 people with blue eyes, and you are one of them. On day one, you see 5 people with blue eyes. Nobody leaves on night one, since everyone knows that there are at least 5 others that have blue eyes. On night two, nobody leaves for the same reason. Same on night three and night four and night five. But, on day six, when you see everyone is still there, that means that each of them see five people with blue eyes, otherwise they all would have left on night 5. So, you realize that there must be six people with blue eyes, and you are the sixth individual. Everyone else also comes to that conclusion, and on night 6, everyone with blue eyes leaves.
@@commander8625 So person 10 knows that no one else knows their eye color, so from their perspective, every other blue eyed person must see 8 other people with blue eyes. And if every "person 9" sees 8 people with blue eyes, they must assume there is 7 people with blue eyes each assuming there is 6, and so on. Makes sense, but I feel like this only works under the assumption that everyone knows each other is a perfect logician.
Because the brown eyed people all know that there's 10-11, while the blue-eyed people know there's 9-10. All the blue-eyed people think that the leaving will happen on Day 9, while all the brown-eyed ones are expecting day 10.
You set up this very unprecisely. There are "many" people on the island. You forgot to specify that every on the island has seen everyone else, or that the stranger has seen them. With your setup the island may as well be Manhattan or Greenland. If a stranger tells the people of Manhattan someone there has blue eyes this will not confirm anything but that someone on Manhattan has blue eyes. Your statement about the stranger telling nothing, and it actually doing something necessitates these further causes, but you've failed to provide them. Simply put, there are few people on the island, though at least 10.
That doesn’t work. The stranger only says “I see someone with blue eyes.” He actually makes no reference to there being 10 blues total. Since no one knows that they have blue eyes, they must see the confirmed blues watch each othwr (think two spider man pointing at each other meme) in anticipation. Since each person sees 9 blues on day 9 do nothing, it must be true that they are the 10th, since they only count 9 blues. Realizing they are the hidden 10th, they leave on night 10. Because every blue doesn’t know they themselves are blue, all 10 head to the boat
If the people on the island know they have to leave if they have blue eyes, I will assume they know how many people there are on the island with blue eyes. With that in mind, all blue eyed people will see the otherd, go "oh I have to be the tenth one" or w/e and then all 10 will leave on day 1.
It's a funny thought to imagine ten people waking up one morning and all simultaneously realizing their own eye color upon observing that the status quo has never changed
The essential bit is not what they hear from the stranger - it's the knowledge they gain when they see no one's left the island the following day
So except for the case where there's only one blue-eyed person (in which their existence is actual information to them) the stranger just acts as a starting signal for the day counting. Which means that if everyone already sees more than one blue-eyed person, they could just agree together on a day that they start counting… actually this would even work for the 1 case, because simply the brown-eyed ppl initiating this agreement would replace the stranger's statement as information for the only person who didn't already know! I guess this would mean allowing this "special meeting" as an exception to the "we don't talk abt eye color" rule, so the stranger's introduction is more natural, but still!
Stranger gives you the knowledge that other people have the knowledge that all of you knew but didn't know that they did. The stranger is essential as he isn't starting an arbitrary countdown, he's converting converting Nth Order Knowledge to Common Knowledge.
It should be pointed out that while the puzzle lets you know there are 10 people with blue eyes, that is not information the islanders have. They don't know how many people have blue eyes. But since they all see at least one person with blue eyes, it appears the stranger's statement doesn't add any new information.
But, in the first walk through where there's only 1 person with blue eyes, the stranger actually does give out information that wasn't known to everyone.
The question really becomes why didn't the people with blue eyes leave the island on their own after 10 days of the rule being in place?
Lol you are right =]] this logic puzzle is so weird
Well... Lets see If there is no stranger in this puzzle, what will happen? In the case there is only 1 person with blueeyes, of course he/she will not leave on his/her own. In the case there are 2 persons with blueeyes, they also will not leave on their own .In the case there are 3 persons, they also will not leave on their own.... All because they are not sure about their eyes color and they know everyone else also doesn't know about their eyes color... This puzzle is so weird =]]
They don't know that at least one person has blue eyes.
What if there's none? No one should leave
If they knew that there was at least one person with blue eyes (that's basically what the stranger told them), they would leave all on say ten
@@someoneunknown6894 They do know. They all see 9 other people with blue eyes.
"Blue eyes" activated my Ted-Ed recognition
Yooo fr tho this comment just activated mine
It wasn't made clear that the islanders do not know how many there are with blue-eyes. Couple of the comments I see here were thrown off by that. It's also probably important to include that if and _only if_ an islander learns they are blue-eyed, they leave the island.
I know there's an effort to save space and words for simplicity sake, but there's also some key details too important to only include in the voice-over and not the text. Specifically, the islanders being forbidden to talk about eye colors and that they have the opportunity to leave (via ferry) _every_ midnight. I think this kind of defeats the purpose of showing text since it looks like everything you need to pause and think about the question, but will just leave the reader confused if they didn't pay attention to the voice-over.
However, I assume that if they didn't have previsous knowledge of the amount of blue-eyed people, the brown-eyed people would think the same, so they would also leave, right? Correct me if I'm wrong btw.
The blue-eyed people will leave 1 day earlier and the brown-eyed people know that they don't have to leave if the blue-eyed people are gone.
If there are 10 blue-eyed people and you are one of them you see 9 blue-eyed people. On day ten the 9 havent left which means (using the logic from the video) there are 10 blue-eyed people and you are the tenth.
On the other hand, if youre brown-eyed you will see 10 blue-eyed people and they will wait until day 11 to see if they leave. And they will leave
I love this because the original has everyone with blue eyes and if every islander had blue eyes i would totally think i did to
wont any brow eyed person will think that maybe they are waiting for him to leave the island as well
they see one more so they would wait one day longer, at which point the blue eyed islanders would have left
@sinisteak6790 ohhhh
@@sinisteak6790 That's a funny way too prank them, the blue eyed people decide not to leave, and every brown eyed person thinks "Oh they must be waiting for me, i must be the last one with blue eyes!" and leaves
@Irondragon1945 that goes against the ground rules tho, everyone has perfect immediate logic and once you know you have blue eyes you must leave
Worst vacation destination ever.
The statement of the puzzle in isolation seems ambiguous to me. There are 10 people with blue eyes, but crucially nobody on the island knows the total number of people with blue eyes - if they did they could simply just count how many people they can see with blue eyes.
It would be more accurate to just say "Everyone on the island has either blue or brown eyes", rather than "10 people have blue eyes, and many have brown eyes".
When you state the puzzle in this way, it clearly shows how the stranger is definitely not "saying nothing" when they say that they can see at least one person with blue eyes because it explicitly rules out the possibility that there are zero people on the island with blue eyes to begin with.
Aside from that it's still an interesting puzzle and a cool inductive solution for any arbitrarily sized island/number of blue-eyed people.
No one knows the exact number of people with blue eyes, but everyone knows that there are at least 1 (or in this case at least 9).
@@mrWade101 My point is that this is not made explicit in the statement of the puzzle
I still have no idea what actual information the stranger brings to the group
I almost broke my brain trying to fit all this in my head, and it's gonna be terribly phrased, but it's basically many levels of abstract assumptions.
Person 1 assumes that they have brown eyes, and there are 9 people with blue eyes. They know that if there are indeed 9 blue-eyed people, Person 2 would have to assume that there are 8 blue-eyed people (of course they know that this isn't the case). Then based on this assumption, to learn their eye colour, Person 3 would have to assume there are 7 blue-eyed people etc etc.
Of course everyone knows that there are 9 other people with blue eyes, but this fact won't get them anywhere. They can only learn if there actually are 9 or 10 blue-eyed people by following this wild chain of assumptions, and disproving them one-by-one. Which includes assuming that someone assumes that someone assumes... that there are no people, which can only be disproven by this stranger's statement.
It's really hard to wrap your head around all this, but it all comes down to ignoring the fact that obviously there aren't fewer than 9 people with blue eyes, and following this train of thought that actually allows you to find out if there are more than 9.
Sorry if this sounds needlessly convoluted.
They aren't allowed to talk about their eye colors. So, everyone knows there's at least 9 blue-eyed people, but as long as no one brings it up, it doesn't matter. So, essentially, they're all pretending they see nothing, but the stranger's words mean they don't have a valid excuse anymore.
POV your the stranger who turns up only to see that everyone already left because they have watched this video...
I paused just before you stated that everyone having perfect logic is common knowledge, and i was JUST about to comment that assumption is EXTREMELY important.
Wait. In the last example with 10 people with blue eyes, does the stranger say that there are 10 people with blue eyes or that there's at least 1 person with blue eyes?
From my understanding, they would say "at least 1", but I don't think it really matters, whatever is the number of blue eyes people, let's say n, if you have blue eyes you see n-1 blue eyes and if there would have been n-1 blue eyes people, they should all leave the n-1th night, and since it's not the case you know that there is actually n blue eyes and you have blue eyes
Ok, the phrase "after the stranger makes the statement, it becomes common knowledge" is driving me completely insane. I think it's a horribly misleading phrasing, probably with a specific meaning to logicians (but if so, it's not adequately explained, along with "9th order knowledge").
Would it be accurate to say that, as perfect logicians, the stranger's statement "starts the clock", so to speak, and everyone knows that once the day counter PASSES the number of blue-eyed people that each blue-eyed person sees (which happens one day sooner than for those with brown eyes), then each blue-eyed person self-identifies?
If so, then why not just say they were all dumped on the island at the same time, with the same rules. The ONE AND ONLY time the stranger matters is in the case of only a single blue-eyed person! In every other case, everyone starts out knowing what they stranger said, so claiming that starts the clock seems completely unnecessary. All you have to do is identify a day 1, and this way seems to be intentionally misleading, in a way that is simple to avoid.
Still, have to give credit for explaining the puzzle well enough that I finally understood it. I know the terminology and set up wasn't chosen by this channel.
Nicely explained!
And all because the stranger could keep quiet
couldn't
Goddamn strangers. Coming to our island. Talking 'bout eyes....
I lost my job cause of that guy!
This is confusing because in the 1 person case, you don't yet know, but in the case with 10 people, you do know
With 1 person, if everyone else has brown eyes, you must be the person with blue eyes that the stranger saw
@@junj1023 Yeah, I'm aware. In the ten person case, everyone already knows there is a person with blue eyes, though. Everyone knows there are at least 9 blue eyed people.
Of course, this knowledge can be expanded on indefinitely. Even if there was a massive space prison with thousands of blue eyed people and the same rules, the single statement from the stranger can eventually release every single blue eyed person from their prison… even though it will take a few years to get to that point.
This makes sense for the examples with 1 or 2 people, but for 3+ people, wouldn't they just go, "Huh, they did not leave. I guess they assumed he was talking about the other guy."
The idea for 3 people is that because everyone is perfectly logical, the two blue eyed people would have left on night two if they looked around and saw only one other blue eyed person. But, since they didn't leave, you learned that there is at least one more blue eyed person on the island, totaling to 3. If you don't see the third person with blue eyes, it must then be you that has blue eyes, so you need to leave too. Therefore, on night 3, all the blue eyed people leave.
The same logic extends past three, just taking an extra night per extra person with blue eyes.
For example, let's run through the logic if there are 6 people with blue eyes, and you are one of them.
On day one, you see 5 people with blue eyes. Nobody leaves on night one, since everyone knows that there are at least 5 others that have blue eyes. On night two, nobody leaves for the same reason. Same on night three and night four and night five. But, on day six, when you see everyone is still there, that means that each of them see five people with blue eyes, otherwise they all would have left on night 5. So, you realize that there must be six people with blue eyes, and you are the sixth individual. Everyone else also comes to that conclusion, and on night 6, everyone with blue eyes leaves.
@@commander8625 So person 10 knows that no one else knows their eye color, so from their perspective, every other blue eyed person must see 8 other people with blue eyes. And if every "person 9" sees 8 people with blue eyes, they must assume there is 7 people with blue eyes each assuming there is 6, and so on.
Makes sense, but I feel like this only works under the assumption that everyone knows each other is a perfect logician.
@3Blue1Brown is this where the channel name comes from??
Why would the brown eyed people not think they are the last blue eyed person? And in turn olso leave?
Because the brown eyed people all know that there's 10-11, while the blue-eyed people know there's 9-10. All the blue-eyed people think that the leaving will happen on Day 9, while all the brown-eyed ones are expecting day 10.
dr sean my goat
You set up this very unprecisely. There are "many" people on the island. You forgot to specify that every on the island has seen everyone else, or that the stranger has seen them. With your setup the island may as well be Manhattan or Greenland. If a stranger tells the people of Manhattan someone there has blue eyes this will not confirm anything but that someone on Manhattan has blue eyes.
Your statement about the stranger telling nothing, and it actually doing something necessitates these further causes, but you've failed to provide them.
Simply put, there are few people on the island, though at least 10.
thats cuz this dude is literally just stealing the TedED riddle, but changing the eye color
I really really enjoyed your video, what an amazing seemingly paradox!
Thanks for this! In future can you use a darker colour for the background? The white is too bright too look at.
TedED puzzle
Neat
Hi, I really liked your video. I just have a question. Do you know your eyes are blue?
or count people with blue eyes and decide if you have blue eyes as well
That doesn’t work. The stranger only says “I see someone with blue eyes.” He actually makes no reference to there being 10 blues total. Since no one knows that they have blue eyes, they must see the confirmed blues watch each othwr (think two spider man pointing at each other meme) in anticipation. Since each person sees 9 blues on day 9 do nothing, it must be true that they are the 10th, since they only count 9 blues. Realizing they are the hidden 10th, they leave on night 10. Because every blue doesn’t know they themselves are blue, all 10 head to the boat
they dont know theres 10 blue eyed people
If the people on the island know they have to leave if they have blue eyes, I will assume they know how many people there are on the island with blue eyes.
With that in mind, all blue eyed people will see the otherd, go "oh I have to be the tenth one" or w/e and then all 10 will leave on day 1.