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Current time:0:00Total duration:11:00

So we had the hundred
logicians. All of their foreheads
were painted blue. And before they entered the
room, they were told that at least one of you hundred
logicians has your forehead painted blue. And then every time that they
turned on the lights, so that they could see each other,
they said OK, once you've determined that you have a blue
forehead, when the lights get turned off again, we want
you to leave the room. And then once that's kind of
settled down, they'll turn the lights on again. And people will look at
each other again. And then they'll turn
them off again. And maybe people will
leave the room. And so forth and so on. And they're also all told that
everyone in the room is a perfect logician. They have infallible logic. So the question was,
what happens? And actually maybe an even more
interesting question is why does it happen? So I'll answer the first,
what happens? And if just take the answer,
and you don't know why, it almost seems mystical. That essentially the light gets
turned on and off 100 times, and then after the
hundredth time that the light gets turned on, and the lights
get turned off again, all of them leave. They all leave.
So I mean, it's kind of weird, right? Let's say I'm one of them. Or you're one of them. I go into this room. The lights get turned on. And I see 99 people with
blue foreheads. And I can't see my
own forehead. They see my forehead,
of course. But to any other person, I'm
one of the 99, right? But I see 99 blue foreheads. So essentially what happens if
we were to watch the show is, the lights get turned on. You see 99 blue foreheads. Then the lights get
turned off again. And then the lights get
turned on again. And everyone's still
sitting there. And I still see 99
blue foreheads. And that happens 100 times. And after the hundredth time
the light gets turned on, everyone leaves the room. And at first glance,
that seems crazy, because nothing changes. Nothing changes between every
time we turn on the light. But the way you need to think
about this-- and this is what makes it interesting-- is what
happens instead of 100, let's say there was one person
in the room. So before the show starts-- they
never told me that there were going to be 100
people in the room. They just said, at least one
of you, at least one of the people in the room, has your
forehead painted blue. And as soon as you know that
your forehead is painted blue, you leave the room. And that everyone's a
perfect logician. So imagine the situation where
instead of 100 there's only one perfect logician. Let's say it's me. So that's the room. I walk in. And I sit down. And maybe I should
do it with blue. And then they turn the
lights on, and say, look around the room. And I look around the room, and
I see nobody else, right? And remember, even in the case
of one, we've painted everyone's forehead blue. So in this case, this one dude,
or me or whoever you want to call him. His forehead is painted blue. So he looks around and he
sees no one in the room. But he remembers the statement,
and maybe it's even written down on a card for
him in case he forgets. That at least one of you has
your forehead painted blue. So if he looks around the room
and he says, well I'm the only dude in the room. And they told me that at least
one of the dudes in the room is going to have their
foreheads blue. Well, I'm the only
dude in the room. So I must have a
blue forehead. So as soon as they turn the
lights off, he's going to leave. Fair enough. That's almost trivially
simple. And you might say, so how
does this apply to 100? Well what happens when
there are two people. And once again, both
of them have their foreheads painted blue. So let me draw another. I don't want to keep drawing
the blue forehead room. Let's say there's
two people now. So let's put ourselves in
the head of this guy. Right behind the
blue forehead. That's where we're sitting. OK. So when he enters the room. He says, I either have
a blue forehead. I either have a blue
forehead, or I don't have a blue forehead. No blue. Right? This is what this
guy's thinking. Let me draw him. And he has a blue forehead. But he doesn't know it. He can't see it. That's the whole point about
painting the forehead blue, as opposed to another
part of the body. So he says, I either have a blue
forehead or I don't have a blue forehead. He walks in. Let's say this is this guy. He walks in. The first time the lights get
turned on, he sees this other dude there who has
a blue forehead. And he says OK, now let
me think about it. How will this guy respond
depending on each of these states? So let's say that I don't
have a blue forehead. Let's go into this reality. If I don't have a blue forehead,
what is this guy going to see? When the lights get turned on,
he's going to see that I don't have a blue forehead. And we were both told
that at least one of us has a blue forehead. So this guy, because he's a
perfect logician, will deduce that he has to have
a blue forehead. Remember, this is in a
situation, if I assume that I don't have a blue forehead. We're in this world. I'm a perfect logician, so
if I can assume, if I'm simulating the reality where I
don't have a blue forehead, then this guy will see, I don't
have a blue forehead. And then he'll say, I must
have a blue forehead. And so when the lights get
turned off, this guy will leave. He'll exit the room. And vice versa. The other guy will make
the same logic. But since both of them have blue
foreheads, what happens? The lights get turned
off, then the lights get turned back on. When the light gets turned
back on, this guy's still sitting here. And I just determined that
if I didn't have a blue forehead-- and this is me--
this guy would have left. Because he could've said, oh I
must be the only guy with a blue forehead. So he would have left. But since he didn't leave,
I now know that I have a blue forehead. So then the second time that
the lights get turned on, I can deduce that I have
a blue forehead. And then when the lights get
turned off again, I'll leave. And this guy, he's a perfect
logician, so he makes the exact same conclusion. Because he also simulated in his
head, OK if I didn't have a blue forehead, then this guy
will leave as soon as the lights get turned off
the first time. If this guy doesn't have a blue
forehead, then this guy will say, well I see no-one else
in the room with a blue forehead, and since I know that
there's at least one with a blue forehead, I'll have to
leave. But since he didn't leave, this guy will also know
that he must have a blue forehead as well, so they'll
actually leave together. Maybe they'll bump into each
other on the way out. Fair enough. Now what happens if you extend
it to three people? So we already said, if you have
one person in the room, he'll come to the conclusion the
first time that the lights are turned on. And then he'll leave right
when they're turned off. If you have two people, it takes
them essentially two times for the light to get
turned on to reach that conclusion. Now if you have three people,
and I think you see where this is going. One, two and three. Now remember, no one knows if
their foreheads are painted blue, but the producers of the
show actually did paint everyone's forehead blue. So so once again, let's get
into this guy's head. So this guy says, he's either
blue or he's not blue. So in the reality when
he's not blue, what's going to happen? Well, this guy-- and this gets a
little bit confusing, but if you think about it from the
previous example, it makes a lot of sense. A person who has a not-blue
forehead actually shouldn't affect the outcome of what
the blue people do. Because let's say that this guy
says, well what's going to happen if I'm not blue? Well, this guy's going to look
at that guy, and say, oh he has a blue forehead. He doesn't have a
blue forehead. So if I don't have a blue
forehead, this guy's going to see two people without
blue foreheads. And he's going to leave the room
the first time that the lights are turned on. He'll come to the conclusion. Now the second time that the
light's turned on, this guy will say, gee, this guy
didn't leave the room. That guy doesn't have
a blue forehead. And this guy didn't leave the
room because he must have seen someone with a blue forehead. Therefore I must have
a blue forehead. And so, if this guy doesn't have
a blue forehead, both of these guys would leave the room
the second time that the light gets turned on. Now what happens if they don't
leave the room the second time that the lights are turned on? Well if I was not blue,
they would have left. So if they haven't left by the
third showing of the light, then I know that I'm blue. So when you have three people,
they're all perfect logicians, they all have their foreheads
painted blue. The light will be shown
three times. Or the light will be turned
on three times. And then when the light
gets turned off, they'll all leave together. And so this logic
applies for any. You could have 1,000 people. You can keep extending it. The fourth person will have
the exact same logic. If he's not blue, then these
guys are going to leave after three turnings on
of the light. But if they don't leave after
three turnings on of the light, then he must be blue. And so they're all going
to leave together. All four of them on
the fourth showing of everyone's foreheads. Anyway, and you can keep
extending this all the way to 100. And 100 is arbitrary. You could do this with a million
people, and they would just keep looking at each
other a million times. And then on the millionth
showing, they would all reach the conclusion that they all
have blue foreheads, and they would leave the room. And if you think about
it, it's fairly straightforward logic. But it leads to kind of a very
almost eerie result. Hopefully that satisfies you. See you in the next video.