There's a new reality television program, and it's
called the Blue-- I should probably write it in blue--
but it's called the Blue Forehead Room.
And what they do in this reality television program--
and you'll have to bear with me, because the show probably
wouldn't be that interesting to watch-- but it's
interesting to predict what happens.
What they do is, they take a room.
They'll call it the blue forehead room.
And let's see, that's kind of a top view of the room.
And let's say there's a door here.
None of this is relevant to the actual problem.
This is the door, right there.
And what they do is they get 100 perfect logicians to sit
in this room, in a circle.
So they're all sitting in a circle in this room.
Now, before the game even starts, before they even enter
the room the first time, the logicians are told two things.
They're told, One: that at least one of you has your
forehead painted blue.
At least one of you has your forehead-- And they all get
their foreheads painted, so that obviously if you're the
only guy who has your forehead painted.
But you just don't know what color it is.
So all of them have different color foreheads.
Or, we don't know.
But all they're told is, obviously I've
painted your forehead.
At least one of the people in the room that you will enter
will have their forehead painted blue.
And then they're also told that as soon as you deduce
that your forehead is blue, you need to leave the room.
And what's going to happen is-- and it's very important
that I set this up properly.
They're all outside of the room.
No-one's inside the room.
And let's say they're blindfolded.
And while they're blindfolded, they essentially have the
thing painted onto their forehead.
So they can't see the paintbrush or anything.
So they really don't know what's on their forehead.
And then after that, they all enter the room.
And they all sit in a circle like this, so that they can
all look at each other.
And let's say when they enter, the lights are off.
So the lights are off, and then the protocol is that the
lights will be turned on, and then they can all look around
at each other.
There's no reflective surfaces.
They can't look into each other's eyes and try to see
the reflection.
No tricks like that.
There are no mirrors in this room.
Nothing like that.
All they can do is look at each other.
So, just as an example, let's say that
this is me right here.
As soon as the lights get turned on the first time, I'll
be able to look at all the other people in the room.
And I could see, it'll be pretty obvious to me if anyone
has a blue forehead.
Maybe that guy has one, that guy doesn't.
I don't know, right?
And I can see them.
I can't see my own forehead.
And what happens is, then they will turn off the lights, and
the way they're going to do it is you have to leave the room
after you have realized that you have a blue forehead.
So for example, let's say I enter into the room, and
because I'm a perfect logician I see things that allow me to
perfectly deduce that I have a blue forehead.
Then what they're going to do is they're going to turn off
the lights again.
And then, if I know that I have a blue forehead while the
lights are turned off, I would leave the room.
And then when they turn the lights back on, I'd be gone.
So there would be no Sal here.
So let's say there were 100 before, then there would be 99
guys sitting in the room.
Right?
As soon as I realize I have a blue forehead, when the lights
get turned off, I leave. And just remember these are
perfect logicians.
So everyone in the room.
And not only are they all perfect logicians, but they
all know that everyone else is a perfect logician.
So, everyone is also told, and this is true, everyone is a
perfect logician.
Which means they have infallible powers of logic.
So my question to you-- Just remember, I have each of these
perfect logicians.
We set them up outside of the room, paint their foreheads.
They're blindfolded.
They have no clue.
Then we have all of them walk into a dark room, sit in a
circle like this.
And then what we tell them is, as soon as you realize that
you have a blue forehead, as soon as you have a blue
forehead, you have to leave the room.
Now my question to you is, let's say that we've actually
painted everybody's forehead blue.
What happens?
So remember, this is what we've told each of the people.
Right?
As soon as you realize your forehead is blue,
you leave the room.
And now I've just asked you, the producers like to really
play with these logicians.
They've actually painted everybody's forehead blue.
So when everyone goes in the room the first time, what's
going to happen?
Let's say I'm one of the logicians.
This is me right here.
As soon as I open my eyes, I'm going to see 99 other fellow
logicians with blue foreheads.
And then, maybe I can somehow deduce something about my own.
I don't think you can.
And the lights will go off.
And then if I haven't deduced anything about my own blueness
of my forehead, then they'll turn the lights back on.
And then maybe some other guy will have left.
I don't know.
Or maybe not.
And then I'll see the same 99 guys again.
And that'll just keep occurring
until something happens.
And my question to you is what happens?
When?
And why?
I'm thinking whether I should give you a hint right now.
Well let me tell it to you this way.
That's the problem.
You should be able to solve it.
And just so you know where this came from, if I remember
correctly I think this was on a computer science
exam I had at MIT.
So just so you know, this isn't just for fun.
I don't want to go into all of the applications that this
type of problem can apply to, because that by itself would
be a bit of a hint.
So if you don't want a hint, turn off the video
now or pause it.
If you do want a hint, I'm about to give it to you.
So my hint is-- and remember turn this off if you don't
want to hint-- but the hint is what happens when there are
less than 100 people.
So I gave the situation where we have 100 logicians.
But this problem is actually a lot easier if you try it with
a smaller number.
Anyway I'll see you in the solution video.