
Blue Eyes Logic Puzzle - ZeljkoS
http://www.math.ucla.edu/~tao/blue.html
======
Mindless2112
Edit: it looks like I'm wrong.

The first argument is true; the second has the logical flaw. The flaw is
assuming that induction can continue despite additional pre-knowledge
available when there are greater numbers of blue-eyed people.

The statement will have no effect when the number of blue-eyed people is 3 or
more:

When the number of blue-eyed people is 0, the foreigner is lying, and if the
tribe believes him, everyone commits ritual suicide.

When the number of blue-eyed people is 1, the blue-eyed person did not know
there were any blue-eyed people in the tribe. Knowledge is added by the
statement, and the blue-eyed person commits ritual suicide.

When the number of blue-eyed people is 2, the blue-eyed people knew that there
was a blue-eyed person but did not know that the blue-eyed person knew that
the tribe had any blue-eyed people. Knowledge is added by the statement, and
the blue-eyed people commit ritual suicide.

When the number of blue-eyed people is 3 or more, the blue-eyed people knew
that there were blue-eyed people and knew that the blue-eyed people knew that
there were blue-eyed people. No knowledge is added by the statement, and no
one commits ritual suicide.

Edit: clarity.

~~~
MereInterest
If n=3, then each of the blue-eyed people know that there are blue-eyed people
and know that the other blue-eyed people know. However, they don't know that
all the blue-eyed people know that the blue-eyed people know. This is the
piece of information that is learned by the statement given.

In general, if there are N blue-eyed people, then it is the Nth abstraction of
"he knows that I know that he knows that I know that..." that is learned by
the statement.

~~~
voyou
"they don't know that all the blue-eyed people know that the blue-eyed people
know"

Yes, they do. In the three-person case, a blue-eyed person can see two other
blue-eyed people, A and B, and they know that A can see B, and vice-versa, so
they know that A and B both know that there are blue-eyed people, and they
know that both A and B would be able to us the same logic they used, so they
also know that A and B know that A and B know that there are blue-eyed people.

------
DanielStraight
Randall Munroe has a much more thorough write-up on (a variation on) this
puzzle:

[http://xkcd.com/solution.html](http://xkcd.com/solution.html) (Though you
might want to click his link to the problem description first since his is a
variation.)

~~~
md224
And for an even more detailed examination of the underlying philosophical
issues:

[http://en.wikipedia.org/wiki/Common_knowledge_(logic)](http://en.wikipedia.org/wiki/Common_knowledge_\(logic\))

[http://plato.stanford.edu/entries/common-
knowledge/](http://plato.stanford.edu/entries/common-knowledge/)

~~~
bonobo
I understand the induction steps, but what I don't get is why the foreigner's
statement triggers the logic induction. This quote from your first link sums
it well:

    
    
        What's most interesting about this scenario is that, for k > 1, the outsider
        is only telling the island citizens what they already know: that there
        are blue-eyed people among them. However, before this fact is announced,
        the fact is not common knowledge.
    

It seems natural to me why they didn't commit the suicide before the statement
(somehow induction doesn't work here), and why they did it after the
statement, but I don't understand why. Isn't the fact that there are k > 1
islanders with blue eyes _common knowledge_ too?

I mean, what bit of information is added here?

~~~
matchu
What's added is the common knowledge that everyone _else_ knows that Blue > 1
(including the foreigner), and that everyone else knows that everyone else
knows that Blue > 1, etc.

Consider two blue-eyed people, Alice and Bob. Alice sees Bob's blue eyes and
knows that Blue ≥ 1, and vice-versa. But Alice thinks, "What if I have brown
eyes? In that case, Bob wouldn't know that Blue ≥ 1." So, everyone knows that
Blue ≥ 1, but nobody knows that everyone knows that. Then the foreigner comes
along and tells them that, between Alice and Bob, Blue ≥ 1. Now Alice knows
that Bob knows that Blue ≥ 1, and realizes that, if Alice has non-blue eyes,
Bob will use the new information that Blue ≥ 1 to conclude that he has blue
eyes, and therefore commit suicide on the next day. When he doesn't, she
concludes that she must have blue eyes, and commits suicide. Bob goes through
the exact same logic as Alice.

Though the foreigner's statement did not tell Alice anything new about eye
color distribution, it _did_ tell her something about Bob's knowledge. The
same goes for Bob, who learns about Alice's knowledge.

The logic is a bit more difficult to talk through with three people, so
generalizing it further is left as an exercise ;P It comes down to "everyone
knows that everyone knows that Blue ≥ 1", which the foreigner _also_
contributes as common knowledge by making his announcement. For more blue-eyed
people, recur on that statement as many times as necessary.

~~~
matchu
Small clarification: For the two-person case, it's important that everyone
knows that everyone knows that Blue ≥ 1. For the three-person case, it's
important that everyone knows that everyone knows that everyone knows that
Blue ≥ 1. (The last paragraph is a bit unclear and/or wrong.)

------
hornd
I think the foreigners statement is ambiguous enough to render the proof in
argument 2 incorrect. "... another blue-eyed person", to me, implies a
singular person in the tribe has blue eyes. All of tribespeople will see
multiple tribespeople with blue eyes, and therefore assume the foreigners
statement was wrong, rendering no effect.

~~~
Mindless2112
It's not so much that the statement was ambiguous or wrong -- it's that the
statement gave no one in the tribe more information than was previously
available to him/her.

Everyone in the tribe already observes at least 99 members with blue eyes, and
everyone knows that everyone else observes at least 99 members with blue eyes,
so the statement should have no effect.

Edit: split to separate comment.

~~~
dllthomas
But they don't know "everyone knew on day X" so there is ambiguity about when
they would be counting from, so nothing can be deduced. The foreigner's
statement provides a synchronization point: clearly everyone knew at that
point.

------
dnautics
> Now suppose inductively that n is larger than 1. _Each blue-eyed person will
> reason as follows:_ “If I am not blue-eyed, then there will only be n-1
> blue-eyed people on this island, and so they will all commit suicide n-1
> days after the traveler’s address”.

Why should not a brown-eyed person reason as follows as well? It is at this
stage that an implicit "counting" of the blue-eyed population creeps into the
flawed proof.

EDIT: I misidentified the place where the flaw comes in. Will repost a better
explanation.

~~~
tbrake
They would. The traveller has accidentally doomed them all.

edit: this is based on an unfounded assumption because of the wording of the
problem - they may only know they don't have blue eyes. But when I read "100
have blue eyes and 900 have brown" it makes it sound binary and I assumed
that's knowledge the tribes people have as well, i.e. we have only either blue
or brown eyes.

~~~
thedufer
In Randall Munroe's version of the problem, he gives the visitor red eyes to
explicitly avoid this binary assumption. Obviously, the visitor's statement is
slightly different because of this.

------
aolol
Taking the 'dramatic effect' argument further: if all the blue-eyed people
kill themselves, would the brown-eyed people all simultaneously know they have
brown eyes and also have to kill themselves?

~~~
Jehar
This is the missing element from most explanations I see of this problem. We
all look at the n cases from the POV of a blue-eyed person. An outside
observer with brown eyes has the same level of information available to him,
so it seems to me just as likely that after the first day, each person,
regardless of eye color, could reason that nobody left the previous day, so
the visitor must have been referring to me. So either eventually everyone
dies, or they all realize the paradox and forgo the ritual.

~~~
Mindless2112
Each blue-eyed person observes 99 other blue-eyed people in the tribe, thus
reasoning that he/she has blue eyes on the 100th day. However, each brown-eyed
person observes 100 blue-eyed people in the tribe, thus reasoning that he/she
has blue eyes on the 101st day (however this does not happen because on the
100th day all the blue-eyed people commit ritual suicide.)

~~~
IanDrake
Doesn't that presuppose they know there are 100 blue eyed people in their
tribe? When that information is presented to the reader, it's presented as
outside knowledge.

If they knew the color counts, they would know their eye color and all would
have to commit suicide. The fact that the tribe still existed means, they
didn't know the totals.

For example, if I know there are 100 people with blue eyes and I can count as
many without including myself, then I must have brown eyes and must kill
myself.

So again, there is no possible way the tribe had any idea what the _exact_
counts where.

As a brown eyed person, there are either 100 blue eyed people meaning I have
brown eyes, _or_ there are a 101 blue eyed people and I have blue eyes. If a
census was ever taken and the exact number known everyone would have to commit
suicide.

Since the visitor didn't mention an exact number then there is still no way to
know if you have blue or brown eyes.

However, the tribe now knows that the visitor knows he himself has blue eyes.
Will they make him follow their ritual?

Update: OK, after reading the link in the first comment, I _get it_.

~~~
Mindless2112
It is not necessary for the blue-eyed people to know the total number of blue-
eyed people in the tribe, they can deduce it at day 100:

    
    
      * A blue-eyed person observes 99 blue-eyed people.
      * On day 99, the blue-eyed people do not commit ritual
        suicide.
      * Thus each blue-eyed person learns that all the blue-eyed
        people also observe 99 blue-eyed people.
      * Thus the blue-eyed person knows that the other blue-eyed
        people must observe that he/she has blue eyes.

------
vytasgd
The traveler has no effect. The logical flaw is that with n > 2 blue eyed
people, everybody knows that there is at least 1 blue eyed person AND
everybody knows that everybody ELSE knows that there is at least 1 blue eyed
person.

The traveler's comments would only have an effect with n<=2 blue eyed people.
With n = 1, he'd instantly know. With n = 2, the 2nd blue eyed person would
recognize that the first person now has the information and if he doesn't
commit suicide on the first night, then the 2nd blue eyed person knows the 1st
blue had the information before, meaning he saw somebody else, and then they
both die on night 2. n > 2, the info is already out that blue-eyed ppl exist
and the count has already started.

------
Symmetry
So, what the visitor is providing is really the coordination, the point at
which you can measure 100 or 99 days. But doesn't this setup require that
there have always been 100 blue eyed people since forever? Any birth or death
or all the islanders being crated at once would serve equally well as a timer.
It seems like this problem only works because the blue-eyed islanders all know
that there are 99 other islanders with blue eyes, but there was no moment in
time where they learned it. And since that is so contrary to our expectations,
it's what ends up making the whole scenario seem so unintuitive.

~~~
lisper
No, the key is that the foreigner's statement establishes common knowledge at
some point in time. What happened before that point in time is irrelevant.

> the blue-eyed islanders all know that there are 99 other islanders with blue
> eyes

That's true, but what they don't know (until the 2nd) day) is that all the
other blue-eyed islanders know that all the other blue-eyed islanders know
that there are 99 other blue-eyed islanders. On the 3rd day they will realize
that all the other blue-eyed islanders know that all the other blue eyed
islanders know that all the other blue eyed islanders know that... and so on.
Then on the 100th day there are 100 iterations of recursive knowledge, and all
the blue-eyed islanders realize they themselves must have blue eyes.

UPDATE: note that it is crucial that all the islanders are together when the
foreigner makes his statement. If he goes to each islander individually and
says "some of you have blue eyes" then it doesn't work. What matters is not
the statement, but that all the islanders witness all the other islanders
hearing the statement.

~~~
dllthomas
Strictly, all _blue eyed people_ need to hear the statement, right? If someone
is missing and everyone (but them) knows the missing person has brown eyes,
that doesn't change the logic of those who heard.

~~~
lisper
That's right.

Which suggests some follow-on puzzles:

1\. What happens if one blue-eyed person is somewhere else on the island when
the foreigner makes his statement (and his absence is known to everyone)?

2\. What happens if the next day a blue-eyed stranger wanders into the
village, thereby establishing common knowledge that the day before there was
in fact an additional blue-eyed person on the island (though no one in the
village knew it at the time)?

3\. What happens if the next day a blue-eyed baby is born in the village?

~~~
thaumasiotes
1\. Suppose the foreigner makes his statement to a group of islanders _C_
("contaminated"), and the rest of the islanders _P_ ("pure") do not hear it,
and it is known to all that they didn't hear it. Call the group of blue-eyed
people _B_. Then the intersection of _C_ with _B_ will kill themselves after a
number of days equal to the size of that group.

2\. Nothing. (I interpreted this as being without a statement by a foreigner.)

3\. Nothing. (I also interpreted this one as being without a statement by a
foreigner. With such a statement, it's the same problem as case 1; everyone
will recognize that the baby, having not existed on Foreigner Day, can't know
about nor have been mentioned in the statement.)

EDIT:

I should point out that I've assumed the foreigner's statement refers to the
group he's addressing, not to the population of the island. ("At least one of
you who I see before me has blue eyes".)

With a better interpretation of your problem 2:

2a. On some day, the foreigner addresses a village, saying "at least one
person on the island has blue eyes". A blue-eyed stranger wanders into the
village shortly after he leaves, allowing the villagers to believe that he was
referring to the stranger.

In this case, there is no synchronization point, and "nothing" will still
occur.

2b. A blue-eyed stranger wanders into the village _the day after_ the
foreigner leaves, allowing the villagers to believe that he was referring to
the stranger.

As far as I can see, this has gone back to case 1 again. The foreigner's
statement provoked a first day of blue-counting, and while it is revealed to
have possibly not meant what they thought it meant, day 1 of blue-counting is
sufficient for day 2. The blue-eyed villagers should kill themselves after a
number of days equal to the size of their group. (The stranger, even if he
settles into the village, will be unaffected.)

------
thaumasiotes
Here is the part that people seem to miss:

> If a tribesperson does discover his or her own eye color, then their
> religion compels them to commit ritual suicide _at noon the following day_
> in the village square for all to witness.

Emphasis mine, of course.

You can think of this clause as specifying the clock speed (one cycle per day)
of the logical machine that is the island.

------
Smaug123
There's an infuriating variant, which I have as yet been unable to solve:

An infinite sequence of people have either blue or brown eyes. They must shout
out a guess as to their own colour of eyes, simultaneously. Is there a way for
them to do it so that only finitely many of them guess incorrectly?

~~~
anonymoushn
And none of them have any knowledge about anything?

~~~
Smaug123
They can all see everyone else's eyes. That is, person N can see person M's
eyes, for all M,N. [I don't know whether it's possible or not - it feels not,
but it has been hinted to me that it is possible.]

------
dllthomas
Personally, I'm convinced the blue-eyed people die on the hundredth day _if
they haven 't earlier_ \- I am not convinced there is no shorter path to the
information (though I certainly don't know of one).

------
informatimago
1- Nothing says that there were 1000 islanders on the day the alien made his
speech. Some may have discovered they eye color and died before.

2- Compare: “how unusual it is to see another blue-eyed person like myself in
this region of the world” with: “how unusual it is to see other blue-eyed
persons like myself in this region of the world”

To me, it is clear that the entire tribe the day the stranger makes his speech
consists of N brown-eyed, and one single remaining blue-eyed.

They immediately understand the same thing, and all commit suicide the next
noon.

------
konceptz
This is a lovely way to teach induction and proofs as part of a class. I
personally prefer the dragon version of this problem. An elegant solution is
provided here.

[https://www.physics.harvard.edu/uploads/files/undergrad/prob...](https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol2.pdf)

------
lazyant
Shouldn't it be better if people had to commit suicide in their own houses? I
mean if there are 2 people with blue eyes and on the second day they see each
other at the town square about to commit seppuko they'd figure perhaps the
other is the only one with blue eyes. Or this whole thing went over my head.

------
spongerbakula
Ok, I'm pretty sure I'm being moronic and missing something, but does the
foreigner give the tribe any more information? Does the tribe already know how
many blue eyed and brown eyed people there are?

~~~
miahi
If they already knew how many of each, then they would all be dead, as any of
them would see that the numbers add up only if he was in a specific group.

------
DannoHung
I think the problem also is missing some part where the people know for sure
that they can only have blue or brown eyes.

------
elwell
Well let's try the experiment a few times and see if our results match up with
our theories in double-blind tests.

~~~
thaumasiotes
The problem statement specifies that the population of the island all reason
with perfect logic. Compare that to the reasoning displayed by my babysitter's
son once:

    
    
        Me: I saw my parents wrapping a Christmas present, but on Christmas
            when I received that present, it was labeled "from Santa".
    
        Him: Santa Claus is real.
    

Good luck finding a suitable population to experiment on. ;)

