Toggler brain teaser

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Kind: captions
Language: en

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This brain teaser in this video
is once again not one

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that I made up.

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I actually don't remember the
first time that I heard it.

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I think it was in the
back of a magazine.

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I want to say Scientific
American.

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And me and a bunch of
buddies, we were

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traveling through Europe.

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And it kind of became our
pastime to do these brain

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teasers in the back
of magazines.

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And just so you know, I think a
lot of you think that these

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type of brain teasers, you can
just read them and solve them.

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And if you can't solve them,
then you're not good at this

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type of thing.

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This brain teaser I'm about to
give you, I just kind of sat

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and thought about it, and me
and my buddies, we argued

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about it for literally
about a day.

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And then I just slept on it.

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And then first thing in the
morning, I woke up all my

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friends, I was like,
I figured it out!

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So this isn't some kind
of easy thing.

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And I encourage you, just
listen to the statement,

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understand it, and then
pause the video.

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And then think about
it for 24 hours.

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And then come back to it.

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One, you'll probably
solve it yourself.

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And if you don't, in 24 or 48
hours, then you'll probably

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find the solution that
much more satisfying.

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So this is the problem
of the togglers.

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Not toddlers.

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Togglers.

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I have an infant, so words like
toddler are in my brain.

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But this is the problem
of the togglers.

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So there are five guys
in front of you.

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That's one of them.

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Two.

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Three.

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Four.

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Five.

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They don't have to be guys.

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They can be of any gender.

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And four of them are what
we call togglers.

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So four of them.

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Four togglers.

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And what a toggler does is, the
first time you ask them a

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question, they're either going
to tell the truth or they're

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going to lie.

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So truth or a lie.

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But then the second time
that you ask them,

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they're going to toggle.

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They're going to switch.

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So the second time you ask them
a question, if they told

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the truth the first time, then
the next time you ask that

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same person a question,
they're going

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to switch to lying.

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But if they lied the first
time you asked them a

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question, then they're
going to switch

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to telling the truth.

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And then it goes on, and
so on, and so forth.

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So if you ask the same person
the third question, if they

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told the truth the first time,
then they'll lie the second

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time you ask them a question,
and then the third time

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they'll tell the truth again.

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It's like, if I was a toggler
and you said, hey you,

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what is your name?

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And if I said, oh
my name is Bill.

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Then clearly I'm lying
that first time.

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And then you pointed to
me again, and you say,

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what is your name?

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And I say, oh my name is Sal.

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And if you ask me again,
what is your name?

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I would say, oh, my
name is Susan.

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And if you ask me again,
my name is Sal.

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So the first time, I just
decided to lie.

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But just as easily, I could've
decided to tell the truth the

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first time.

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You might have said, hey
you, what's your name?

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I would have said, Sal.

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And then you ask me, what's
your name again?

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And I would have
said Elizabeth.

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And then you ask me, what's
my name again?

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I would say, oh,
my name is Sal.

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And then again, what's
your name?

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And I would say, you know,
George Washington

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or something arbitrary.

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But I think you get the idea.

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A toggler-- I don't know
who's a toggler.

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Let's say that this
is a toggler.

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The first time you ask them a
question, they might lie, they

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might tell the truth.

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But if they lie, the next time
you ask the same person a

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question, they're going
to tell the truth.

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But the first time you ask them
a question, and they did

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tell the truth, the next time
you ask the same person a

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question, he or she
is going to lie.

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So that's a toggler.

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So you can already imagine,
this is-- I don't know.

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The first time that I read the
problem it kind of made my

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brain hurt.

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This notion of a toggler.

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It's much easier when there's
a reality where people only

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lie or only tell the truth.

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It's much harder when
they toggle.

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And not only do they toggle,
but their initial state,

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whether they start telling the
truth or they start off lying.

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You don't know.

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That's arbitrary.

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There's no way of predicting
whether this person-- First of

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all, we don't know if this
person is a toggler, because I

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said there's only
four togglers.

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But even if they are toggler, I
don't even know if the first

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time I ask them a question
whether they're going to tell

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the truth or whether they're
going to lie.

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So four of them are togglers.

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What's the fifth?

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Well, you probably could guess
because this is how these

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brain teasers go.

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The fifth is a truth teller.

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And he is much simpler.

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Or he or she.

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I don't want to give any clues
on who it might be.

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Is a truth teller.

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And a truth teller always
tells the truth.

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So the goal of this brain teaser
is to figure out who is

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the truth teller.

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So you want to know which of
these five people is-- So

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let's say, goal.

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Who's truth teller?

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And as you can imagine, you
don't have infinite questions

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to do this with.

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You have exactly two questions
to do this with.

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And you have two questions.

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And those two questions, they
can be one question to this

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guy and another question
to this guy.

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Both questions could
be to this guy.

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Both questions could be to
her right over there.

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So the questions don't have
to be to the same person,

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although they can be
to the same person.

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Or they could be to two
different people.

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So you have two questions.

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Any question to ask of any
combination of either two

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questions to one person or two
questions to two people.

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And by the end of those two
questions, you should know for

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sure who is the truth teller.

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And that is the brain
teaser statement.

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So pause it now if you don't
want any hints or the

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solution, or anything else.

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So I encourage you to pause it
and think about it for at

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least a day.

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And sleep on it.

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Because a lot of the real math
problems or logic problems

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that are worth solving aren't
the ones that you can solve

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right when you see them.

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They're the ones where you kind
of sit and let your brain

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do a little background
processing while you sleep.

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So that said, hopefully you've
stopped it and now you've

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resumed it after really
struggling with this for a

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long period.

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Not eating and sleeping and
all the things that

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normal people do.

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And now you're ready
for the solution.

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Or even better, you're ready
to confirm your solution.

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So first I'll give you a hint.

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I mean, it's a tough question.

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I struggled with this at first.
I was like, when you

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ask someone a question
how do you know if

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they're lying or not?

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How do you know if they're
the truth teller?

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How do you know if they're
a toggler?

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You don't have any
information.

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So the crux of this problem is,
essentially, it doesn't

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matter who you're asking
the question to

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the first time around.

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And actually the solution is
that it doesn't matter who

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you're asking the question to
the second time around.

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You want to ask a question so
that no matter what they say,

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you know what they're going to
do the second time around.

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You know whether they're going
to tell the truth the second

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time around, or you know whether
they're going to lie

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the second time around.

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And that's a hint.

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And if you just want a hint
and then to keep thinking

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about it, pause it again.

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Because I'm about to give
you the solution.

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So question one.

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And you can pick any of the
five people to ask this

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question to.

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And that's kind of the nature
of one of these solutions.

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There's no reason for you
to pick one of the

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five over the other.

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They're all equally likely
to be the truth teller.

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So you ask them, are you
the truth teller?

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So let's think about a couple
of-- Well first of all,

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they're either going to answer
to this, yes or no.

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So there could be
two situations.

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They're either going to
say yes, or they're

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going to say no.

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So if someone says yes to are
they the truth teller, it

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means that either they
are the truth

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teller telling the truth.

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Or they are a lying toggler.

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If they say that they are not
the truth teller, they clearly

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cannot be the truth teller,
because the truth teller

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cannot lie about not being
the truth teller.

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And so they have to
be a toggler.

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And they'd be an
honest toggler.

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They'd be a toggler who's
telling the truth.

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Right?

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So it would be a truthful
toggler.

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Remember, I ask anybody, any
of these five people right

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here, say, are you
the truth teller?

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If they say yes, I immediately
know that they're either the

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truth teller telling
the truth, or

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they're a lying toggler.

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If they say no, I know that they
are a truthful toggler.

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Now, what's interesting is, in
this scenario, if they say

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yes, what do I know that they're
going to do next?

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So if this is the truth teller,
and they told the

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truth, on the second question,
they're going to tell the

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truth again.

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They're going to tell the truth
on question number two.

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If they're the lying toggler,
what are they going to do when

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you ask them another question?

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Well, they toggle.

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If they lie the first time you
ask them, they're going to

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tell the truth the second
time you ask them.

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So they're going to tell the
truth on question number two.

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So if the person says yes to,
are you the truth teller?

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You immediately know that no
matter who they are, that

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they're going to tell the truth
on question number two.

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So if you can know for a fact
that the person's going to

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tell the truth on question
number two, you ask them, who

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is the truth teller?

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So this is number two.

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Who is the truth teller?

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So it's kind of weird.

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You go to someone and say,
are you the truth teller?

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And they'll say yes.

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And then you ask them again,
who's the truth teller?

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And then, because they have to
tell the truth, they will

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actually either say, if it was
this case, they'd say I am the

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truth teller.

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Or if it's this case, they'd say
no, that dude over there

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is the truth teller.

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And then you'll know who
the truth teller is.

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Now what happens in
this situation?

00:10:01.310 --> 00:10:03.160
That they're a truthful
toggler?

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Well on the second question,
what are they going to do?

00:10:05.800 --> 00:10:08.680
They're going to lie.

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They're going to lie on the
second question, right?

00:10:13.910 --> 00:10:16.310
On question number two.

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So if you know for a fact that
someone's going to lie, how

00:10:19.600 --> 00:10:22.120
can you phrase a question so
you can figure out who the

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truth teller is?

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Well, if they say no to this
first question, and you know

00:10:25.530 --> 00:10:27.430
they're going to lie on the
second one, your second

00:10:27.430 --> 00:10:31.820
question should just be, who
is not the truth teller?

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So the only way to lie on this
question-- and this person has

00:10:41.590 --> 00:10:45.780
to lie now-- is to tell you
who the truth teller is.

00:10:45.780 --> 00:10:49.750
And so, using the first
question, are

00:10:49.750 --> 00:10:50.490
you the truth teller?

00:10:50.490 --> 00:10:52.720
We can immediately predict what
the person's going to do

00:10:52.720 --> 00:10:53.480
on the second question.

00:10:53.480 --> 00:10:56.310
And then we can phrase the
second question, depending on

00:10:56.310 --> 00:11:00.290
that response, to definitely
know who the truth teller is.

00:11:00.290 --> 00:11:01.520
Anyway, I thought you'd
enjoy that.

00:11:01.520 --> 00:11:07.550
And hopefully you see why the
liar game show brain teaser

00:11:07.550 --> 00:11:09.750
was a bit of a warm
up for this one.

00:11:09.750 --> 00:11:11.680
See you in the next video.


Liar truth-teller brain teaser

WEBVTT
Kind: captions
Language: en

00:00:01.030 --> 00:00:04.110
This brain teaser is a bit of
a classic, and I'm sure a

00:00:04.110 --> 00:00:05.600
bunch of you have
heard it before.

00:00:05.600 --> 00:00:08.510
But I wanted to include this in
the brain teaser playlist

00:00:08.510 --> 00:00:09.770
that I'm making right now.

00:00:09.770 --> 00:00:12.480
Because this one is actually a
good build up, and a bit of a

00:00:12.480 --> 00:00:16.379
nice warm up session for a
slightly harder brain teaser

00:00:16.379 --> 00:00:18.840
that deals with people telling
the truth and lying.

00:00:18.840 --> 00:00:25.025
So this one, we could call it
the truth teller and the liar.

00:00:28.350 --> 00:00:31.490
So we're on some type of
adventure quest, and we get to

00:00:31.490 --> 00:00:33.020
a fork in a road, right?

00:00:33.020 --> 00:00:37.680
So let's say that there's
two doors.

00:00:37.680 --> 00:00:38.340
That's even better.

00:00:38.340 --> 00:00:41.710
So there's two doors.

00:00:41.710 --> 00:00:46.770
And I've gotten to the end of my
adventure quest. And I know

00:00:46.770 --> 00:00:50.700
that my treasure is behind
one of these two doors.

00:00:50.700 --> 00:00:57.790
Unfortunately, behind the other
door is slow and painful

00:00:57.790 --> 00:00:59.870
death, and all of the
misery in the world.

00:00:59.870 --> 00:01:02.590
So I want to pick the
door carefully.

00:01:02.590 --> 00:01:06.340
And beside the doors, there are
two people sitting there.

00:01:06.340 --> 00:01:08.780
There's one guy that
looks like this.

00:01:08.780 --> 00:01:11.950
And then there's another guy
that looks like this.

00:01:11.950 --> 00:01:16.270
And I know for a fact - the
gods of brain teasers have

00:01:16.270 --> 00:01:22.250
come and told me - that both of
these guys know which door

00:01:22.250 --> 00:01:27.870
has the treasure, and both of
these guys know which door has

00:01:27.870 --> 00:01:33.350
the unbelievable despair
that would ensue if

00:01:33.350 --> 00:01:34.980
I opened that door.

00:01:34.980 --> 00:01:36.870
So both of these
guys know which

00:01:36.870 --> 00:01:38.270
door I should be opening.

00:01:38.270 --> 00:01:42.070
And I also know, and both of
them know, that one of them

00:01:42.070 --> 00:01:43.740
always tells the truth.

00:01:43.740 --> 00:01:51.325
One is always honest. And then
one of them always lies.

00:01:56.700 --> 00:02:00.370
And the problem is I don't
know who's who.

00:02:00.370 --> 00:02:03.230
And I only have one chance.

00:02:03.230 --> 00:02:04.800
Obviously I can't just open
a door and close it.

00:02:04.800 --> 00:02:06.710
If I open the wrong door,
the despair will

00:02:06.710 --> 00:02:07.990
come out and kill.

00:02:07.990 --> 00:02:09.709
And you know, all of the
pain will happen.

00:02:09.709 --> 00:02:14.030
And I can only ask one question
of one of them to

00:02:14.030 --> 00:02:15.340
decide which door to open.

00:02:15.340 --> 00:02:17.670
So remember, they both know
the correct answer.

00:02:17.670 --> 00:02:19.160
But one of them always lies.

00:02:19.160 --> 00:02:20.710
One of them always
tells the truth.

00:02:20.710 --> 00:02:26.730
And I only have one question
to ask of one of them.

00:02:26.730 --> 00:02:31.350
So the brain teaser is: What
question do you ask?

00:02:31.350 --> 00:02:32.780
Who do you ask it to?

00:02:32.780 --> 00:02:35.700
Do you ask it to the green
guy or to the brown guy?

00:02:35.700 --> 00:02:40.300
And then what do you do based
on what they tell you?

00:02:40.300 --> 00:02:43.370
So that is the problem
description.

00:02:43.370 --> 00:02:46.020
And I will now give
you the solution.

00:02:46.020 --> 00:02:48.150
So if you don't want
the solution, and I

00:02:48.150 --> 00:02:49.110
encourage you not to.

00:02:49.110 --> 00:02:51.320
I think if you think about this
one for maybe 20 or 30

00:02:51.320 --> 00:02:52.690
minutes, or at most a day.

00:02:52.690 --> 00:02:55.020
Sometimes when you sleep
on a brain teaser it

00:02:55.020 --> 00:02:55.770
might come to you.

00:02:55.770 --> 00:02:57.420
So I encourage you to do that.

00:02:57.420 --> 00:02:59.390
So pause it, stop
it, whatever.

00:02:59.390 --> 00:03:00.730
Don't listen anymore.

00:03:00.730 --> 00:03:04.850
Because here is the solution.

00:03:04.850 --> 00:03:07.950
So what you do is-- I mean,
you don't know who's who.

00:03:07.950 --> 00:03:11.170
So it's not like there's a
correct answer on who you ask

00:03:11.170 --> 00:03:12.340
a question of.

00:03:12.340 --> 00:03:20.860
But what you do is you ask you
either of the people, what

00:03:20.860 --> 00:03:31.720
would the other guy say is
the correct door to open?

00:03:35.900 --> 00:03:37.000
So let's think about this.

00:03:37.000 --> 00:03:41.260
So I could ask this guy, I'm
asking the green guy, what

00:03:41.260 --> 00:03:44.220
would this guy say is the
correct door to open?

00:03:44.220 --> 00:03:48.480
Now what happens if this
guy tells the truth?

00:03:48.480 --> 00:03:51.200
If this guy tells the
truth, then this

00:03:51.200 --> 00:03:53.970
guy's the liar, right?

00:03:53.970 --> 00:03:57.790
And so he will actually
tell me what this guy

00:03:57.790 --> 00:03:59.040
would tell me to do.

00:04:04.360 --> 00:04:09.240
So what would the other guy
say is the correct door to

00:04:09.240 --> 00:04:09.960
open, right?

00:04:09.960 --> 00:04:13.950
So this guy, if you were to tell
him the correct door to

00:04:13.950 --> 00:04:16.550
open, he would tell you the
incorrect door to open.

00:04:16.550 --> 00:04:17.440
Right?

00:04:17.440 --> 00:04:22.700
So when you ask this guy what
would that guy say, he'll be

00:04:22.700 --> 00:04:24.590
honest about what this
guy would say.

00:04:24.590 --> 00:04:26.270
And since he's a liar,
he would tell you

00:04:26.270 --> 00:04:27.400
the incorrect door.

00:04:27.400 --> 00:04:29.930
So the answer to this question,
you would get the

00:04:29.930 --> 00:04:31.370
incorrect door.

00:04:31.370 --> 00:04:35.150
Incorrect door would be the
answer to that question if

00:04:35.150 --> 00:04:37.620
this guy's the truth teller
and this guy's the liar.

00:04:37.620 --> 00:04:39.060
Now what happens if it's
the other way around?

00:04:39.060 --> 00:04:42.530
What if I ask this guy and he
happens to be the liar?

00:04:42.530 --> 00:04:47.350
So this guy's the liar, this
guy's the truth teller.

00:04:47.350 --> 00:04:51.210
So the liar, he can't
help himself.

00:04:51.210 --> 00:04:52.570
He will lie.

00:04:52.570 --> 00:04:54.790
So he knows that this other
guy is the truth

00:04:54.790 --> 00:04:55.550
teller there, right?

00:04:55.550 --> 00:04:56.770
There's only two
of them there.

00:04:56.770 --> 00:04:58.400
He knows this guy's
the truth teller.

00:04:58.400 --> 00:05:01.820
And he knows that this guy
when asked will say the

00:05:01.820 --> 00:05:04.000
correct door to open.

00:05:04.000 --> 00:05:06.430
So if you happened to be lucky
and you ask this guy the

00:05:06.430 --> 00:05:09.230
correct door, he would
tell it to you.

00:05:09.230 --> 00:05:13.090
This guy will lie.

00:05:13.090 --> 00:05:17.040
Let's say that this is the
correct door, door number two

00:05:17.040 --> 00:05:18.290
is the correct door.

00:05:22.680 --> 00:05:26.990
So let's say that door number
two is the correct door.

00:05:26.990 --> 00:05:28.565
If you ask this guy directly,
he'll say oh

00:05:28.565 --> 00:05:29.510
yeah, door number two.

00:05:29.510 --> 00:05:32.770
You'll find all your happiness
behind door number two.

00:05:32.770 --> 00:05:36.910
Now if this guy's a liar, he
knows that the truth teller

00:05:36.910 --> 00:05:38.860
will say door number two.

00:05:38.860 --> 00:05:42.030
But he's a liar, so he'll tell
you door number one.

00:05:42.030 --> 00:05:46.130
So once again, if I ask the same
question of the liar, I

00:05:46.130 --> 00:05:49.250
still get the incorrect door.

00:05:49.250 --> 00:05:49.560
Right?

00:05:49.560 --> 00:05:53.350
So no matter who I ask this
question to, their answer will

00:05:53.350 --> 00:05:57.680
be the wrong door, the incorrect
door, or the door

00:05:57.680 --> 00:05:59.800
that I get all of
the pain from.

00:05:59.800 --> 00:06:02.530
So what you do is you ask this
question, what would the other

00:06:02.530 --> 00:06:04.920
guy say is the correct
door to open?

00:06:04.920 --> 00:06:08.150
And no matter who you ask that
to, they're going to answer

00:06:08.150 --> 00:06:09.910
the incorrect door.

00:06:09.910 --> 00:06:12.220
And so you do the opposite.

00:06:12.220 --> 00:06:15.940
So I ask this guy, mister green
guy, what would mister

00:06:15.940 --> 00:06:19.530
brown guy say is the correct
door to open?

00:06:19.530 --> 00:06:26.960
And green guy will say brown guy
will tell you to open door

00:06:26.960 --> 00:06:28.430
number one.

00:06:28.430 --> 00:06:31.280
Well green guy says that brown
guy will tell me to open door

00:06:31.280 --> 00:06:35.040
number one, then I should
open up door number two.

00:06:35.040 --> 00:06:39.290
And likewise, if green guy says
brown guy will say to

00:06:39.290 --> 00:06:43.450
open door number two, then I
should open door number one.

00:06:43.450 --> 00:06:46.680
Anyway that's a nice warm up
brain teaser for the next one

00:06:46.680 --> 00:06:48.390
I'm going to give you.


Blue forehead room solution

WEBVTT
Kind: captions
Language: en

00:00:01.540 --> 00:00:04.500
So we had the hundred
logicians.

00:00:04.500 --> 00:00:06.570
All of their foreheads
were painted blue.

00:00:06.570 --> 00:00:09.910
And before they entered the
room, they were told that at

00:00:09.910 --> 00:00:15.240
least one of you hundred
logicians has your forehead

00:00:15.240 --> 00:00:16.440
painted blue.

00:00:16.440 --> 00:00:18.920
And then every time that they
turned on the lights, so that

00:00:18.920 --> 00:00:21.420
they could see each other,
they said OK, once you've

00:00:21.420 --> 00:00:23.820
determined that you have a blue
forehead, when the lights

00:00:23.820 --> 00:00:27.040
get turned off again, we want
you to leave the room.

00:00:27.040 --> 00:00:30.120
And then once that's kind of
settled down, they'll turn the

00:00:30.120 --> 00:00:31.270
lights on again.

00:00:31.270 --> 00:00:32.659
And people will look at
each other again.

00:00:32.659 --> 00:00:34.820
And then they'll turn
them off again.

00:00:34.820 --> 00:00:35.970
And maybe people will
leave the room.

00:00:35.970 --> 00:00:38.150
And so forth and so on.

00:00:38.150 --> 00:00:40.810
And they're also all told that
everyone in the room is a

00:00:40.810 --> 00:00:41.700
perfect logician.

00:00:41.700 --> 00:00:43.380
They have infallible logic.

00:00:43.380 --> 00:00:45.560
So the question was,
what happens?

00:00:45.560 --> 00:00:48.990
And actually maybe an even more
interesting question is

00:00:48.990 --> 00:00:49.420
why does it happen?

00:00:49.420 --> 00:00:51.880
So I'll answer the first,
what happens?

00:00:51.880 --> 00:00:55.050
And if just take the answer,
and you don't know why, it

00:00:55.050 --> 00:00:56.880
almost seems mystical.

00:00:56.880 --> 00:01:02.520
That essentially the light gets
turned on and off 100

00:01:02.520 --> 00:01:05.640
times, and then after the
hundredth time that the light

00:01:05.640 --> 00:01:15.540
gets turned on, and the lights
get turned off again, all of

00:01:15.540 --> 00:01:21.350
them leave. They all leave.
So I mean, it's

00:01:21.350 --> 00:01:23.730
kind of weird, right?

00:01:23.730 --> 00:01:24.660
Let's say I'm one of them.

00:01:24.660 --> 00:01:25.750
Or you're one of them.

00:01:25.750 --> 00:01:27.060
I go into this room.

00:01:27.060 --> 00:01:28.570
The lights get turned on.

00:01:28.570 --> 00:01:32.090
And I see 99 people with
blue foreheads.

00:01:32.090 --> 00:01:35.020
And I can't see my
own forehead.

00:01:35.020 --> 00:01:36.520
They see my forehead,
of course.

00:01:36.520 --> 00:01:40.810
But to any other person, I'm
one of the 99, right?

00:01:40.810 --> 00:01:42.690
But I see 99 blue foreheads.

00:01:42.690 --> 00:01:45.970
So essentially what happens if
we were to watch the show is,

00:01:45.970 --> 00:01:47.390
the lights get turned on.

00:01:47.390 --> 00:01:49.150
You see 99 blue foreheads.

00:01:49.150 --> 00:01:50.910
Then the lights get
turned off again.

00:01:50.910 --> 00:01:53.970
And then the lights get
turned on again.

00:01:53.970 --> 00:01:57.030
And everyone's still
sitting there.

00:01:57.030 --> 00:01:59.700
And I still see 99
blue foreheads.

00:01:59.700 --> 00:02:02.490
And that happens 100 times.

00:02:02.490 --> 00:02:05.510
And after the hundredth time
the light gets turned on,

00:02:05.510 --> 00:02:06.810
everyone leaves the room.

00:02:06.810 --> 00:02:13.230
And at first glance,
that seems crazy,

00:02:13.230 --> 00:02:15.210
because nothing changes.

00:02:15.210 --> 00:02:21.010
Nothing changes between every
time we turn on the light.

00:02:21.010 --> 00:02:24.030
But the way you need to think
about this-- and this is what

00:02:24.030 --> 00:02:26.380
makes it interesting-- is what
happens instead of 100, let's

00:02:26.380 --> 00:02:29.070
say there was one person
in the room.

00:02:29.070 --> 00:02:32.485
So before the show starts-- they
never told me that there

00:02:32.485 --> 00:02:33.890
were going to be 100
people in the room.

00:02:33.890 --> 00:02:36.010
They just said, at least one
of you, at least one of the

00:02:36.010 --> 00:02:39.310
people in the room, has your
forehead painted blue.

00:02:39.310 --> 00:02:41.410
And as soon as you know that
your forehead is painted blue,

00:02:41.410 --> 00:02:42.920
you leave the room.

00:02:42.920 --> 00:02:44.950
And that everyone's a
perfect logician.

00:02:44.950 --> 00:02:48.430
So imagine the situation where
instead of 100 there's only

00:02:48.430 --> 00:02:49.500
one perfect logician.

00:02:49.500 --> 00:02:52.100
Let's say it's me.

00:02:52.100 --> 00:02:55.570
So that's the room.

00:02:55.570 --> 00:02:57.460
I walk in.

00:02:57.460 --> 00:02:58.760
And I sit down.

00:02:58.760 --> 00:03:02.160
And maybe I should
do it with blue.

00:03:02.160 --> 00:03:04.540
And then they turn the
lights on, and say,

00:03:04.540 --> 00:03:06.310
look around the room.

00:03:06.310 --> 00:03:08.890
And I look around the room, and
I see nobody else, right?

00:03:08.890 --> 00:03:11.525
And remember, even in the case
of one, we've painted

00:03:11.525 --> 00:03:12.540
everyone's forehead blue.

00:03:12.540 --> 00:03:15.360
So in this case, this one dude,
or me or whoever you

00:03:15.360 --> 00:03:16.270
want to call him.

00:03:16.270 --> 00:03:18.110
His forehead is painted blue.

00:03:18.110 --> 00:03:21.970
So he looks around and he
sees no one in the room.

00:03:21.970 --> 00:03:23.880
But he remembers the statement,
and maybe it's even

00:03:23.880 --> 00:03:26.830
written down on a card for
him in case he forgets.

00:03:26.830 --> 00:03:30.310
That at least one of you has
your forehead painted blue.

00:03:30.310 --> 00:03:32.180
So if he looks around the room
and he says, well I'm the only

00:03:32.180 --> 00:03:33.850
dude in the room.

00:03:33.850 --> 00:03:36.470
And they told me that at least
one of the dudes in the room

00:03:36.470 --> 00:03:38.110
is going to have their
foreheads blue.

00:03:38.110 --> 00:03:39.870
Well, I'm the only
dude in the room.

00:03:39.870 --> 00:03:41.620
So I must have a
blue forehead.

00:03:41.620 --> 00:03:43.550
So as soon as they turn the
lights off, he's going to

00:03:43.550 --> 00:03:46.450
leave. Fair enough.

00:03:46.450 --> 00:03:49.010
That's almost trivially
simple.

00:03:49.010 --> 00:03:52.040
And you might say, so how
does this apply to 100?

00:03:52.040 --> 00:03:53.820
Well what happens when
there are two people.

00:03:53.820 --> 00:03:56.870
And once again, both
of them have their

00:03:56.870 --> 00:03:57.810
foreheads painted blue.

00:03:57.810 --> 00:03:59.480
So let me draw another.

00:03:59.480 --> 00:04:02.060
I don't want to keep drawing
the blue forehead room.

00:04:02.060 --> 00:04:06.130
Let's say there's
two people now.

00:04:06.130 --> 00:04:12.400
So let's put ourselves in
the head of this guy.

00:04:12.400 --> 00:04:13.650
Right behind the
blue forehead.

00:04:13.650 --> 00:04:15.270
That's where we're sitting.

00:04:15.270 --> 00:04:15.620
OK.

00:04:15.620 --> 00:04:19.660
So when he enters the room.

00:04:19.660 --> 00:04:24.380
He says, I either have
a blue forehead.

00:04:28.712 --> 00:04:31.010
I either have a blue
forehead, or I

00:04:31.010 --> 00:04:33.220
don't have a blue forehead.

00:04:33.220 --> 00:04:35.490
No blue.

00:04:35.490 --> 00:04:36.180
Right?

00:04:36.180 --> 00:04:39.230
This is what this
guy's thinking.

00:04:39.230 --> 00:04:40.050
Let me draw him.

00:04:40.050 --> 00:04:41.900
And he has a blue forehead.

00:04:41.900 --> 00:04:42.590
But he doesn't know it.

00:04:42.590 --> 00:04:43.180
He can't see it.

00:04:43.180 --> 00:04:46.050
That's the whole point about
painting the forehead blue, as

00:04:46.050 --> 00:04:48.170
opposed to another
part of the body.

00:04:48.170 --> 00:04:50.890
So he says, I either have a blue
forehead or I don't have

00:04:50.890 --> 00:04:52.440
a blue forehead.

00:04:52.440 --> 00:04:55.800
He walks in.

00:04:55.800 --> 00:04:56.950
Let's say this is this guy.

00:04:56.950 --> 00:04:57.850
He walks in.

00:04:57.850 --> 00:05:00.930
The first time the lights get
turned on, he sees this other

00:05:00.930 --> 00:05:03.410
dude there who has
a blue forehead.

00:05:03.410 --> 00:05:05.480
And he says OK, now let
me think about it.

00:05:05.480 --> 00:05:08.900
How will this guy respond
depending on

00:05:08.900 --> 00:05:10.220
each of these states?

00:05:10.220 --> 00:05:13.070
So let's say that I don't
have a blue forehead.

00:05:13.070 --> 00:05:14.730
Let's go into this reality.

00:05:14.730 --> 00:05:17.900
If I don't have a blue forehead,
what is this guy

00:05:17.900 --> 00:05:19.180
going to see?

00:05:19.180 --> 00:05:22.620
When the lights get turned on,
he's going to see that I don't

00:05:22.620 --> 00:05:23.760
have a blue forehead.

00:05:23.760 --> 00:05:25.770
And we were both told
that at least one of

00:05:25.770 --> 00:05:27.440
us has a blue forehead.

00:05:27.440 --> 00:05:31.070
So this guy, because he's a
perfect logician, will deduce

00:05:31.070 --> 00:05:32.690
that he has to have
a blue forehead.

00:05:32.690 --> 00:05:35.630
Remember, this is in a
situation, if I assume that I

00:05:35.630 --> 00:05:36.720
don't have a blue forehead.

00:05:36.720 --> 00:05:38.700
We're in this world.

00:05:38.700 --> 00:05:43.920
I'm a perfect logician, so
if I can assume, if I'm

00:05:43.920 --> 00:05:46.690
simulating the reality where I
don't have a blue forehead,

00:05:46.690 --> 00:05:49.550
then this guy will see, I don't
have a blue forehead.

00:05:49.550 --> 00:05:51.950
And then he'll say, I must
have a blue forehead.

00:05:51.950 --> 00:05:55.260
And so when the lights get
turned off, this guy will

00:05:55.260 --> 00:05:57.950
leave. He'll exit the room.

00:05:57.950 --> 00:05:59.680
And vice versa.

00:05:59.680 --> 00:06:01.770
The other guy will make
the same logic.

00:06:01.770 --> 00:06:05.940
But since both of them have blue
foreheads, what happens?

00:06:05.940 --> 00:06:07.680
The lights get turned
off, then the lights

00:06:07.680 --> 00:06:09.020
get turned back on.

00:06:09.020 --> 00:06:11.270
When the light gets turned
back on, this guy's still

00:06:11.270 --> 00:06:13.490
sitting here.

00:06:13.490 --> 00:06:15.780
And I just determined that
if I didn't have a blue

00:06:15.780 --> 00:06:22.790
forehead-- and this is me--
this guy would have left.

00:06:22.790 --> 00:06:26.920
Because he could've said, oh I
must be the only guy with a

00:06:26.920 --> 00:06:28.000
blue forehead.

00:06:28.000 --> 00:06:29.710
So he would have left.

00:06:29.710 --> 00:06:32.630
But since he didn't leave,
I now know that

00:06:32.630 --> 00:06:34.300
I have a blue forehead.

00:06:34.300 --> 00:06:40.420
So then the second time that
the lights get turned on, I

00:06:40.420 --> 00:06:42.030
can deduce that I have
a blue forehead.

00:06:42.030 --> 00:06:44.500
And then when the lights get
turned off again, I'll leave.

00:06:44.500 --> 00:06:46.950
And this guy, he's a perfect
logician, so he makes the

00:06:46.950 --> 00:06:49.460
exact same conclusion.

00:06:49.460 --> 00:06:52.420
Because he also simulated in his
head, OK if I didn't have

00:06:52.420 --> 00:06:59.790
a blue forehead, then this guy
will leave as soon as the

00:06:59.790 --> 00:07:02.330
lights get turned off
the first time.

00:07:02.330 --> 00:07:06.140
If this guy doesn't have a blue
forehead, then this guy

00:07:06.140 --> 00:07:11.390
will say, well I see no-one else
in the room with a blue

00:07:11.390 --> 00:07:13.870
forehead, and since I know that
there's at least one with

00:07:13.870 --> 00:07:15.990
a blue forehead, I'll have to
leave. But since he didn't

00:07:15.990 --> 00:07:18.200
leave, this guy will also know
that he must have a blue

00:07:18.200 --> 00:07:20.000
forehead as well, so they'll
actually leave together.

00:07:20.000 --> 00:07:24.250
Maybe they'll bump into each
other on the way out.

00:07:24.250 --> 00:07:25.520
Fair enough.

00:07:25.520 --> 00:07:29.390
Now what happens if you extend
it to three people?

00:07:29.390 --> 00:07:32.570
So we already said, if you have
one person in the room,

00:07:32.570 --> 00:07:37.060
he'll come to the conclusion the
first time that the lights

00:07:37.060 --> 00:07:37.650
are turned on.

00:07:37.650 --> 00:07:39.390
And then he'll leave right
when they're turned off.

00:07:39.390 --> 00:07:41.780
If you have two people, it takes
them essentially two

00:07:41.780 --> 00:07:43.860
times for the light to get
turned on to reach that

00:07:43.860 --> 00:07:45.020
conclusion.

00:07:45.020 --> 00:07:49.990
Now if you have three people,
and I think you see

00:07:49.990 --> 00:07:51.240
where this is going.

00:07:55.420 --> 00:07:58.240
One, two and three.

00:07:58.240 --> 00:08:00.780
Now remember, no one knows if
their foreheads are painted

00:08:00.780 --> 00:08:03.430
blue, but the producers of the
show actually did paint

00:08:03.430 --> 00:08:05.340
everyone's forehead blue.

00:08:05.340 --> 00:08:10.090
So so once again, let's get
into this guy's head.

00:08:10.090 --> 00:08:14.120
So this guy says, he's either
blue or he's not blue.

00:08:16.990 --> 00:08:20.810
So in the reality when
he's not blue,

00:08:20.810 --> 00:08:22.840
what's going to happen?

00:08:22.840 --> 00:08:26.820
Well, this guy-- and this gets a
little bit confusing, but if

00:08:26.820 --> 00:08:28.790
you think about it from the
previous example, it makes a

00:08:28.790 --> 00:08:30.040
lot of sense.

00:08:33.250 --> 00:08:36.690
A person who has a not-blue
forehead actually shouldn't

00:08:36.690 --> 00:08:39.730
affect the outcome of what
the blue people do.

00:08:39.730 --> 00:08:42.340
Because let's say that this guy
says, well what's going to

00:08:42.340 --> 00:08:43.700
happen if I'm not blue?

00:08:43.700 --> 00:08:48.020
Well, this guy's going to look
at that guy, and say, oh he

00:08:48.020 --> 00:08:49.720
has a blue forehead.

00:08:49.720 --> 00:08:51.820
He doesn't have a
blue forehead.

00:08:51.820 --> 00:08:55.050
So if I don't have a blue
forehead, this guy's going to

00:08:55.050 --> 00:08:58.360
see two people without
blue foreheads.

00:08:58.360 --> 00:09:04.760
And he's going to leave the room
the first time that the

00:09:04.760 --> 00:09:06.650
lights are turned on.

00:09:06.650 --> 00:09:08.670
He'll come to the conclusion.

00:09:08.670 --> 00:09:11.080
Now the second time that the
light's turned on, this guy

00:09:11.080 --> 00:09:14.550
will say, gee, this guy
didn't leave the room.

00:09:14.550 --> 00:09:16.250
That guy doesn't have
a blue forehead.

00:09:16.250 --> 00:09:18.500
And this guy didn't leave the
room because he must have seen

00:09:18.500 --> 00:09:20.150
someone with a blue forehead.

00:09:20.150 --> 00:09:22.540
Therefore I must have
a blue forehead.

00:09:22.540 --> 00:09:25.810
And so, if this guy doesn't have
a blue forehead, both of

00:09:25.810 --> 00:09:28.990
these guys would leave the room
the second time that the

00:09:28.990 --> 00:09:30.580
light gets turned on.

00:09:30.580 --> 00:09:33.050
Now what happens if they don't
leave the room the second time

00:09:33.050 --> 00:09:34.800
that the lights are turned on?

00:09:34.800 --> 00:09:37.000
Well if I was not blue,
they would have left.

00:09:37.000 --> 00:09:40.430
So if they haven't left by the
third showing of the light,

00:09:40.430 --> 00:09:42.660
then I know that I'm blue.

00:09:42.660 --> 00:09:45.270
So when you have three people,
they're all perfect logicians,

00:09:45.270 --> 00:09:48.810
they all have their foreheads
painted blue.

00:09:48.810 --> 00:09:50.860
The light will be shown
three times.

00:09:50.860 --> 00:09:52.950
Or the light will be turned
on three times.

00:09:52.950 --> 00:09:54.800
And then when the light
gets turned off,

00:09:54.800 --> 00:09:56.050
they'll all leave together.

00:09:58.970 --> 00:10:00.760
And so this logic
applies for any.

00:10:00.760 --> 00:10:02.230
You could have 1,000 people.

00:10:02.230 --> 00:10:03.250
You can keep extending it.

00:10:03.250 --> 00:10:08.030
The fourth person will have
the exact same logic.

00:10:08.030 --> 00:10:16.450
If he's not blue, then these
guys are going to leave after

00:10:16.450 --> 00:10:20.430
three turnings on
of the light.

00:10:20.430 --> 00:10:22.670
But if they don't leave after
three turnings on of the

00:10:22.670 --> 00:10:25.030
light, then he must be blue.

00:10:25.030 --> 00:10:26.630
And so they're all going
to leave together.

00:10:26.630 --> 00:10:29.390
All four of them on
the fourth showing

00:10:29.390 --> 00:10:31.340
of everyone's foreheads.

00:10:31.340 --> 00:10:32.860
Anyway, and you can keep
extending this

00:10:32.860 --> 00:10:33.640
all the way to 100.

00:10:33.640 --> 00:10:34.625
And 100 is arbitrary.

00:10:34.625 --> 00:10:36.520
You could do this with a million
people, and they would

00:10:36.520 --> 00:10:39.060
just keep looking at each
other a million times.

00:10:39.060 --> 00:10:41.020
And then on the millionth
showing, they would all reach

00:10:41.020 --> 00:10:43.236
the conclusion that they all
have blue foreheads, and they

00:10:43.236 --> 00:10:44.880
would leave the room.

00:10:44.880 --> 00:10:48.170
And if you think about
it, it's fairly

00:10:48.170 --> 00:10:49.100
straightforward logic.

00:10:49.100 --> 00:10:56.230
But it leads to kind of a very
almost eerie result.

00:10:56.230 --> 00:10:57.700
Hopefully that satisfies you.

00:10:57.700 --> 00:10:59.540
See you in the next video.


Blue forehead room brain teaser

WEBVTT
Kind: captions
Language: en

00:00:01.100 --> 00:00:04.710
There's a new reality television
program, and it's

00:00:04.710 --> 00:00:08.410
called the Blue-- I should
probably write it in blue--

00:00:08.410 --> 00:00:10.570
but it's called the Blue
Forehead Room.

00:00:17.140 --> 00:00:20.480
And what they do in this reality
television program--

00:00:20.480 --> 00:00:22.730
and you'll have to bear with me,
because the show probably

00:00:22.730 --> 00:00:24.650
wouldn't be that interesting
to watch-- but it's

00:00:24.650 --> 00:00:26.190
interesting to predict
what happens.

00:00:26.190 --> 00:00:28.670
What they do is, they
take a room.

00:00:28.670 --> 00:00:31.930
They'll call it the blue
forehead room.

00:00:31.930 --> 00:00:35.695
And let's see, that's kind of
a top view of the room.

00:00:35.695 --> 00:00:37.300
And let's say there's
a door here.

00:00:37.300 --> 00:00:39.140
None of this is relevant
to the actual problem.

00:00:39.140 --> 00:00:42.880
This is the door, right there.

00:00:42.880 --> 00:00:48.050
And what they do is they get 100
perfect logicians to sit

00:00:48.050 --> 00:00:49.220
in this room, in a circle.

00:00:49.220 --> 00:00:56.490
So they're all sitting in
a circle in this room.

00:00:56.490 --> 00:00:59.470
Now, before the game even
starts, before they even enter

00:00:59.470 --> 00:01:04.629
the room the first time, the
logicians are told two things.

00:01:04.629 --> 00:01:15.910
They're told, One: that at
least one of you has your

00:01:15.910 --> 00:01:17.320
forehead painted blue.

00:01:27.030 --> 00:01:29.780
At least one of you has your
forehead-- And they all get

00:01:29.780 --> 00:01:32.600
their foreheads painted, so that
obviously if you're the

00:01:32.600 --> 00:01:34.810
only guy who has your
forehead painted.

00:01:34.810 --> 00:01:36.170
But you just don't know
what color it is.

00:01:36.170 --> 00:01:39.680
So all of them have different
color foreheads.

00:01:39.680 --> 00:01:40.760
Or, we don't know.

00:01:40.760 --> 00:01:43.230
But all they're told
is, obviously I've

00:01:43.230 --> 00:01:44.370
painted your forehead.

00:01:44.370 --> 00:01:47.540
At least one of the people in
the room that you will enter

00:01:47.540 --> 00:01:49.280
will have their forehead
painted blue.

00:01:49.280 --> 00:01:56.500
And then they're also told that
as soon as you deduce

00:01:56.500 --> 00:02:04.980
that your forehead is blue, you
need to leave the room.

00:02:08.430 --> 00:02:10.880
And what's going to happen is--
and it's very important

00:02:10.880 --> 00:02:13.910
that I set this up properly.

00:02:13.910 --> 00:02:15.680
They're all outside
of the room.

00:02:15.680 --> 00:02:17.090
No-one's inside the room.

00:02:17.090 --> 00:02:18.770
And let's say they're
blindfolded.

00:02:18.770 --> 00:02:21.860
And while they're blindfolded,
they essentially have the

00:02:21.860 --> 00:02:25.400
thing painted onto
their forehead.

00:02:25.400 --> 00:02:26.780
So they can't see the paintbrush
or anything.

00:02:26.780 --> 00:02:30.640
So they really don't know what's
on their forehead.

00:02:30.640 --> 00:02:36.330
And then after that, they
all enter the room.

00:02:36.330 --> 00:02:38.720
And they all sit in a circle
like this, so that they can

00:02:38.720 --> 00:02:41.625
all look at each other.

00:02:41.625 --> 00:02:46.000
And let's say when they enter,
the lights are off.

00:02:46.000 --> 00:02:51.180
So the lights are off, and then
the protocol is that the

00:02:51.180 --> 00:02:54.610
lights will be turned on, and
then they can all look around

00:02:54.610 --> 00:02:55.030
at each other.

00:02:55.030 --> 00:02:56.540
There's no reflective
surfaces.

00:02:56.540 --> 00:02:59.400
They can't look into each
other's eyes and try to see

00:02:59.400 --> 00:02:59.970
the reflection.

00:02:59.970 --> 00:03:00.810
No tricks like that.

00:03:00.810 --> 00:03:02.030
There are no mirrors
in this room.

00:03:02.030 --> 00:03:02.730
Nothing like that.

00:03:02.730 --> 00:03:04.440
All they can do is look
at each other.

00:03:04.440 --> 00:03:07.180
So, just as an example,
let's say that

00:03:07.180 --> 00:03:08.860
this is me right here.

00:03:08.860 --> 00:03:11.080
As soon as the lights get turned
on the first time, I'll

00:03:11.080 --> 00:03:13.290
be able to look at all the
other people in the room.

00:03:13.290 --> 00:03:16.190
And I could see, it'll be pretty
obvious to me if anyone

00:03:16.190 --> 00:03:18.100
has a blue forehead.

00:03:18.100 --> 00:03:19.550
Maybe that guy has one,
that guy doesn't.

00:03:19.550 --> 00:03:20.650
I don't know, right?

00:03:20.650 --> 00:03:22.170
And I can see them.

00:03:22.170 --> 00:03:24.160
I can't see my own forehead.

00:03:24.160 --> 00:03:29.460
And what happens is, then they
will turn off the lights, and

00:03:29.460 --> 00:03:32.090
the way they're going to do it
is you have to leave the room

00:03:32.090 --> 00:03:35.290
after you have realized that
you have a blue forehead.

00:03:35.290 --> 00:03:39.550
So for example, let's say I
enter into the room, and

00:03:39.550 --> 00:03:43.530
because I'm a perfect logician
I see things that allow me to

00:03:43.530 --> 00:03:47.890
perfectly deduce that I
have a blue forehead.

00:03:47.890 --> 00:03:49.820
Then what they're going to do
is they're going to turn off

00:03:49.820 --> 00:03:51.240
the lights again.

00:03:51.240 --> 00:03:53.900
And then, if I know that I have
a blue forehead while the

00:03:53.900 --> 00:03:57.080
lights are turned off, I
would leave the room.

00:03:57.080 --> 00:04:02.030
And then when they turn the
lights back on, I'd be gone.

00:04:02.030 --> 00:04:03.450
So there would be no Sal here.

00:04:03.450 --> 00:04:06.450
So let's say there were 100
before, then there would be 99

00:04:06.450 --> 00:04:07.960
guys sitting in the room.

00:04:07.960 --> 00:04:08.200
Right?

00:04:08.200 --> 00:04:12.090
As soon as I realize I have a
blue forehead, when the lights

00:04:12.090 --> 00:04:15.740
get turned off, I leave. And
just remember these are

00:04:15.740 --> 00:04:16.570
perfect logicians.

00:04:16.570 --> 00:04:17.990
So everyone in the room.

00:04:17.990 --> 00:04:19.910
And not only are they all
perfect logicians, but they

00:04:19.910 --> 00:04:22.830
all know that everyone else
is a perfect logician.

00:04:22.830 --> 00:04:33.120
So, everyone is also told, and
this is true, everyone is a

00:04:33.120 --> 00:04:35.440
perfect logician.

00:04:35.440 --> 00:04:38.285
Which means they have infallible
powers of logic.

00:04:44.310 --> 00:04:48.620
So my question to you-- Just
remember, I have each of these

00:04:48.620 --> 00:04:50.030
perfect logicians.

00:04:50.030 --> 00:04:52.820
We set them up outside of the
room, paint their foreheads.

00:04:52.820 --> 00:04:53.590
They're blindfolded.

00:04:53.590 --> 00:04:54.840
They have no clue.

00:04:54.840 --> 00:04:58.010
Then we have all of them walk
into a dark room, sit in a

00:04:58.010 --> 00:04:59.760
circle like this.

00:04:59.760 --> 00:05:03.240
And then what we tell them is,
as soon as you realize that

00:05:03.240 --> 00:05:06.420
you have a blue forehead, as
soon as you have a blue

00:05:06.420 --> 00:05:09.080
forehead, you have to
leave the room.

00:05:09.080 --> 00:05:12.810
Now my question to you is, let's
say that we've actually

00:05:12.810 --> 00:05:16.580
painted everybody's
forehead blue.

00:05:16.580 --> 00:05:18.130
What happens?

00:05:18.130 --> 00:05:21.650
So remember, this is what we've
told each of the people.

00:05:21.650 --> 00:05:21.930
Right?

00:05:21.930 --> 00:05:24.030
As soon as you realize your
forehead is blue,

00:05:24.030 --> 00:05:25.400
you leave the room.

00:05:25.400 --> 00:05:29.460
And now I've just asked you, the
producers like to really

00:05:29.460 --> 00:05:30.740
play with these logicians.

00:05:30.740 --> 00:05:33.150
They've actually painted
everybody's forehead blue.

00:05:33.150 --> 00:05:36.090
So when everyone goes in the
room the first time, what's

00:05:36.090 --> 00:05:38.900
going to happen?

00:05:38.900 --> 00:05:40.400
Let's say I'm one of
the logicians.

00:05:40.400 --> 00:05:42.690
This is me right here.

00:05:42.690 --> 00:05:47.990
As soon as I open my eyes, I'm
going to see 99 other fellow

00:05:47.990 --> 00:05:51.680
logicians with blue foreheads.

00:05:51.680 --> 00:05:56.550
And then, maybe I can somehow
deduce something about my own.

00:05:56.550 --> 00:05:58.480
I don't think you can.

00:05:58.480 --> 00:06:00.340
And the lights will go off.

00:06:00.340 --> 00:06:03.950
And then if I haven't deduced
anything about my own blueness

00:06:03.950 --> 00:06:06.770
of my forehead, then they'll
turn the lights back on.

00:06:06.770 --> 00:06:09.980
And then maybe some other
guy will have left.

00:06:09.980 --> 00:06:11.320
I don't know.

00:06:11.320 --> 00:06:12.080
Or maybe not.

00:06:12.080 --> 00:06:14.120
And then I'll see the
same 99 guys again.

00:06:14.120 --> 00:06:15.530
And that'll just keep occurring

00:06:15.530 --> 00:06:17.630
until something happens.

00:06:17.630 --> 00:06:20.600
And my question to you
is what happens?

00:06:20.600 --> 00:06:21.610
When?

00:06:21.610 --> 00:06:22.860
And why?

00:06:27.380 --> 00:06:29.760
I'm thinking whether I should
give you a hint right now.

00:06:29.760 --> 00:06:30.740
Well let me tell it
to you this way.

00:06:30.740 --> 00:06:31.650
That's the problem.

00:06:31.650 --> 00:06:32.800
You should be able
to solve it.

00:06:32.800 --> 00:06:34.810
And just so you know where this
came from, if I remember

00:06:34.810 --> 00:06:36.740
correctly I think this was
on a computer science

00:06:36.740 --> 00:06:38.150
exam I had at MIT.

00:06:38.150 --> 00:06:41.040
So just so you know, this
isn't just for fun.

00:06:44.000 --> 00:06:48.860
I don't want to go into all of
the applications that this

00:06:48.860 --> 00:06:51.390
type of problem can apply to,
because that by itself would

00:06:51.390 --> 00:06:53.620
be a bit of a hint.

00:06:53.620 --> 00:06:56.110
So if you don't want a hint,
turn off the video

00:06:56.110 --> 00:06:58.090
now or pause it.

00:06:58.090 --> 00:07:00.990
If you do want a hint, I'm
about to give it to you.

00:07:00.990 --> 00:07:05.440
So my hint is-- and remember
turn this off if you don't

00:07:05.440 --> 00:07:11.560
want to hint-- but the hint is
what happens when there are

00:07:11.560 --> 00:07:12.720
less than 100 people.

00:07:12.720 --> 00:07:16.080
So I gave the situation where
we have 100 logicians.

00:07:16.080 --> 00:07:19.050
But this problem is actually a
lot easier if you try it with

00:07:19.050 --> 00:07:20.560
a smaller number.

00:07:20.560 --> 00:07:23.990
Anyway I'll see you in
the solution video.


Solving 3 Equations with 3 Unknowns

WEBVTT
Kind: captions
Language: en

00:00:00.930 --> 00:00:03.320
You're probably pretty familiar
at this point with

00:00:03.320 --> 00:00:06.360
how to solve two equations
and two unknowns.

00:00:06.360 --> 00:00:13.910
Let's say we had the equation
2x, I don't know, minus y is

00:00:13.910 --> 00:00:15.740
equal to 10.

00:00:15.740 --> 00:00:23.270
And let's say you had x
plus 3y is equal to 5.

00:00:23.270 --> 00:00:25.130
And there are a lot of different
ways to solve it.

00:00:25.130 --> 00:00:28.210
And visually, the representation
we use in our

00:00:28.210 --> 00:00:30.260
heads, is that this represents
a line, and this

00:00:30.260 --> 00:00:31.320
represents a line.

00:00:31.320 --> 00:00:33.300
And if they're not parallel to
each other, they're going to

00:00:33.300 --> 00:00:35.280
intersect at a unique point.

00:00:35.280 --> 00:00:38.080
And that unique point will
be the solution.

00:00:38.080 --> 00:00:40.670
That unique x and y will be
the point where these two

00:00:40.670 --> 00:00:41.570
lines intersect.

00:00:41.570 --> 00:00:45.230
And so one way that we can solve
these is we can multiply

00:00:45.230 --> 00:00:48.550
either of these equations
by scalars, by numbers.

00:00:48.550 --> 00:00:50.570
And we have to multiply both
sides of the equation, and

00:00:50.570 --> 00:00:52.370
then add them to the
other equation.

00:00:52.370 --> 00:00:54.280
So what we could do, and we do
it in such a way is that we

00:00:54.280 --> 00:00:56.100
eliminate variables.

00:00:56.100 --> 00:00:59.420
So for example, let's
say we wanted to get

00:00:59.420 --> 00:01:00.680
rid of the y term.

00:01:00.680 --> 00:01:09.655
So we could take this equation
and then add it to what?

00:01:09.655 --> 00:01:12.340
So we have a 3y here.

00:01:12.340 --> 00:01:15.650
So then if we made this into a
minus 3y, when you add the

00:01:15.650 --> 00:01:18.910
two, you would get 0.

00:01:18.910 --> 00:01:20.160
The y's would eliminate.

00:01:22.510 --> 00:01:24.280
Let me do it this way.

00:01:24.280 --> 00:01:29.990
If you multiply this top
equation by 3, you get 6x

00:01:29.990 --> 00:01:32.570
minus 3y is equal to 30.

00:01:32.570 --> 00:01:35.270
Multiply both sides by 3.

00:01:35.270 --> 00:01:37.710
And the bottom equation we're
keeping the same.

00:01:37.710 --> 00:01:42.570
x plus 3y is equal to 5.

00:01:42.570 --> 00:01:44.510
And then you just add the two.

00:01:44.510 --> 00:01:47.830
So 7x.

00:01:47.830 --> 00:01:50.660
That's a 0, is equal to 35.

00:01:50.660 --> 00:01:52.800
I picked these numbers at
random, but it seems that they

00:01:52.800 --> 00:01:53.950
actually work out.

00:01:53.950 --> 00:01:56.540
So we get x is equal to 5.

00:01:56.540 --> 00:01:58.710
And then you could go back, and
you take this, and you can

00:01:58.710 --> 00:02:02.100
substitute it back into either
one of these equations.

00:02:02.100 --> 00:02:04.200
So you would have-- let's
do the first one.

00:02:04.200 --> 00:02:09.229
2 times 5 minus y
is equal to 10.

00:02:09.229 --> 00:02:12.110
10 minus y is equal to 10.

00:02:12.110 --> 00:02:13.800
y would be equal to 10, right?

00:02:13.800 --> 00:02:15.670
10 minus 0 is 10.

00:02:15.670 --> 00:02:18.190
And it's true here in this
equation as well.

00:02:18.190 --> 00:02:21.090
So that's how you solve two
equations of two unknowns.

00:02:21.090 --> 00:02:22.820
So how do you do it with
three equations?

00:02:25.650 --> 00:02:27.190
So I have here-- let
me write it down.

00:02:27.190 --> 00:02:30.800
I designed this equation, so
it's actually reasonably

00:02:30.800 --> 00:02:32.595
straightforward to solve.

00:02:32.595 --> 00:02:44.620
You have minus x plus 2y
minus z is equal to 9.

00:02:44.620 --> 00:02:57.630
I have 3x minus 7y minus 2z
is equal to minus 20.

00:02:57.630 --> 00:03:08.740
And I have 2x plus 2y plus
z is equal to 2.

00:03:08.740 --> 00:03:12.250
Now first of all, what's the
visual representation of this?

00:03:12.250 --> 00:03:15.000
And if you watched the intuition
when I talked about

00:03:15.000 --> 00:03:18.850
the video on the intuition of a
three variable integration,

00:03:18.850 --> 00:03:22.090
you know that each of these
represent a plane in three

00:03:22.090 --> 00:03:23.040
dimensions.

00:03:23.040 --> 00:03:27.410
And so, if you're trying to
figure out what x, y, and z

00:03:27.410 --> 00:03:30.320
satisfy all three of these
equations, you're essentially

00:03:30.320 --> 00:03:34.800
saying, what point in
three-dimensional space exists

00:03:34.800 --> 00:03:37.350
on all three of these planes?

00:03:37.350 --> 00:03:40.080
And that point is essentially
the intersection of these

00:03:40.080 --> 00:03:40.700
three planes.

00:03:40.700 --> 00:03:42.800
And that's very hard to draw,
so I won't do it now.

00:03:42.800 --> 00:03:45.170
But you might want to watch
that visual intuition--

00:03:45.170 --> 00:03:47.380
intuition video that I did.

00:03:47.380 --> 00:03:49.740
But anyway, let's go to
actually solving it.

00:03:49.740 --> 00:03:54.140
So solving three equations and
three unknowns, you actually

00:03:54.140 --> 00:03:55.580
do the same thing as
you did with two

00:03:55.580 --> 00:03:56.550
equations and two unknowns.

00:03:56.550 --> 00:03:58.150
It's just a lot messier.

00:03:58.150 --> 00:04:00.870
And it's just a little bit more
confusing in terms of

00:04:00.870 --> 00:04:03.500
which variable do you need
to eliminate, and when.

00:04:03.500 --> 00:04:06.650
And because of that, it's very,
very, very important to

00:04:06.650 --> 00:04:08.560
be systematic about it.

00:04:08.560 --> 00:04:12.730
So the way I like to do it is,
I like to-- and this is

00:04:12.730 --> 00:04:15.340
actually called Gaussian
elimination, or Gauss-Jordan

00:04:15.340 --> 00:04:18.640
elimination, and it's done a
little bit more formally in

00:04:18.640 --> 00:04:20.170
linear algebra.

00:04:20.170 --> 00:04:24.440
But essentially, I like to
eliminate the two x's in these

00:04:24.440 --> 00:04:26.820
two equations, and then
I like to eliminate

00:04:26.820 --> 00:04:28.530
this y in this equation.

00:04:28.530 --> 00:04:29.950
So all you're left
with is a z here.

00:04:29.950 --> 00:04:31.620
And then you'd have z
equals something.

00:04:31.620 --> 00:04:33.580
Then you can substitute into
this equation, then you can

00:04:33.580 --> 00:04:34.550
substitute for y.

00:04:34.550 --> 00:04:36.700
And then you could substitute
the y and the z back to this

00:04:36.700 --> 00:04:38.150
equation, and solve for x.

00:04:38.150 --> 00:04:38.940
So let's do that.

00:04:38.940 --> 00:04:41.790
I know that sounds a little
daunting, but I think if we

00:04:41.790 --> 00:04:43.600
actually do the problem, it'll
be a little bit more

00:04:43.600 --> 00:04:45.490
straightforward.

00:04:45.490 --> 00:04:48.230
So before we break into the
mechanics, I want to show you

00:04:48.230 --> 00:04:48.860
something else.

00:04:48.860 --> 00:04:51.680
And this is-- depending when
you're doing this, your

00:04:51.680 --> 00:04:55.230
teacher might be upset or
impressed that you do this--

00:04:55.230 --> 00:04:57.980
but this is just an easier
way to save time.

00:04:57.980 --> 00:05:03.980
It's to write an augmented
matrix that represents these

00:05:03.980 --> 00:05:05.400
three equations of
three unknowns.

00:05:05.400 --> 00:05:08.070
And what do I mean by that?

00:05:08.070 --> 00:05:10.985
We're going to be rewriting
these over and over again.

00:05:10.985 --> 00:05:19.590
And instead of writing an x, a
y, a z, and the equal sign

00:05:19.590 --> 00:05:21.470
every time, let's just write
the numbers that matter.

00:05:21.470 --> 00:05:24.800
So we can represent this as-- So
minus x, we could say this

00:05:24.800 --> 00:05:29.690
is minus 1, 2, minus 1.

00:05:29.690 --> 00:05:33.370
And this is called an
augmented matrix.

00:05:33.370 --> 00:05:34.470
Because we'll put a 9 here.

00:05:34.470 --> 00:05:39.660
So this just says, minus x plus
2y minus z is equal to 9.

00:05:39.660 --> 00:05:48.680
And then we say 3x minus 7y
minus 2z is equal to minus 20.

00:05:48.680 --> 00:05:49.720
I'm not doing anything here.

00:05:49.720 --> 00:05:52.100
I'm just rewriting it in a form
that has a little bit

00:05:52.100 --> 00:05:55.150
less letters in it.

00:05:55.150 --> 00:05:59.400
And then the last equation
is 2x plus 2y plus

00:05:59.400 --> 00:06:00.860
z is equal to 2.

00:06:03.960 --> 00:06:06.020
So the first thing I said that
wanted to do, is I wanted to

00:06:06.020 --> 00:06:10.910
eliminate these two
x terms. Right?

00:06:10.910 --> 00:06:15.000
Remember, these just correspond
to these two terms.

00:06:15.000 --> 00:06:18.560
So how do I do that?

00:06:18.560 --> 00:06:22.900
Well I'm going to keep the top
equation the same, minus x,

00:06:22.900 --> 00:06:28.910
plus 2y minus z is equal
to positive 9.

00:06:28.910 --> 00:06:30.160
That's just that equation.

00:06:33.130 --> 00:06:34.670
And remember, you could
write it out.

00:06:34.670 --> 00:06:37.050
You could rewrite these with
x's and y's all the time.

00:06:37.050 --> 00:06:38.870
I'm just doing this for a little
bit of simplicity.

00:06:38.870 --> 00:06:41.050
And your teacher actually
might do it this way.

00:06:41.050 --> 00:06:42.010
So that's more useful.

00:06:42.010 --> 00:06:44.430
And I like to do it this way.

00:06:44.430 --> 00:06:48.910
So how can we eliminate this 3x
here, or this 3 right here?

00:06:48.910 --> 00:06:54.470
Well what if we took this
equation, and added it to 3

00:06:54.470 --> 00:06:55.830
times this equation?

00:06:55.830 --> 00:07:00.370
Or another way, we substituted
this equation with this

00:07:00.370 --> 00:07:04.360
equation, plus 3 times
this equation.

00:07:04.360 --> 00:07:06.250
So what happens?

00:07:06.250 --> 00:07:11.910
So we take each of these terms,
and add to it 3 times

00:07:11.910 --> 00:07:17.040
each of these terms. So 3x plus
3 times minus x, that is

00:07:17.040 --> 00:07:19.356
equal to 0.

00:07:19.356 --> 00:07:20.840
Right?

00:07:20.840 --> 00:07:23.550
I'm just multiplying this top
equation by 3, and adding it

00:07:23.550 --> 00:07:24.830
to this equation.

00:07:24.830 --> 00:07:29.130
So minus 3x plus 3x-- Let me
actually do it one time, just

00:07:29.130 --> 00:07:30.030
so you see what I'm doing.

00:07:30.030 --> 00:07:32.660
I just multiplied this
top equation by 3.

00:07:32.660 --> 00:07:41.980
So you get minus 3x plus 6y
minus 3z is equal to 27.

00:07:41.980 --> 00:07:44.050
And I'm adding that to this.

00:07:44.050 --> 00:07:47.080
So minus 3x plus 3x is 0.

00:07:47.080 --> 00:07:53.350
6y minus 7y is equal
to minus y.

00:07:53.350 --> 00:08:06.650
And then minus 3z plus minus
2z is equal to minus 5z.

00:08:06.650 --> 00:08:09.030
This is very computationally
intensive, you can tell.

00:08:09.030 --> 00:08:13.340
And then 27 plus minus 20,
that's just equal to 7.

00:08:16.770 --> 00:08:18.640
I'm going to dwell on this step
a little bit, because I

00:08:18.640 --> 00:08:19.900
really want you to understand
what I'm doing.

00:08:19.900 --> 00:08:21.480
Because if you understood what
I just did in this step,

00:08:21.480 --> 00:08:24.420
you'll understand the rest
of this video, frankly.

00:08:24.420 --> 00:08:27.670
So I just replaced this
equation-- which is the same

00:08:27.670 --> 00:08:30.780
thing as this equation--
with this equation

00:08:30.780 --> 00:08:32.059
plus 3 times this.

00:08:32.059 --> 00:08:36.260
So I said, 3 plus 3 times
negative 3 is 0.

00:08:36.260 --> 00:08:40.140
Minus 7 plus 3 times
2 is minus 1.

00:08:40.140 --> 00:08:44.950
Minus 2 plus 3 times
minus 1 is minus 5.

00:08:44.950 --> 00:08:48.610
Minus 20 plus 3 times 9 is 7.

00:08:48.610 --> 00:08:49.860
Hopefully that makes sense.

00:08:49.860 --> 00:08:51.190
So now, let's do the
same thing here.

00:08:51.190 --> 00:08:53.350
Let's try to eliminate
this 2x.

00:08:53.350 --> 00:08:55.300
So if you want to eliminate this
2x, we could take this

00:08:55.300 --> 00:08:59.220
equation and add to it 2
times this equation.

00:08:59.220 --> 00:09:00.830
So what will happen?

00:09:00.830 --> 00:09:06.040
We'll have 2 plus
2 times minus 1.

00:09:06.040 --> 00:09:07.910
That's 0.

00:09:07.910 --> 00:09:10.740
2 plus 2 times 2.

00:09:10.740 --> 00:09:12.800
Well that's 6.

00:09:12.800 --> 00:09:18.040
And then we have 1 plus
2 times minus 1.

00:09:18.040 --> 00:09:20.020
Well that's minus 1.

00:09:20.020 --> 00:09:20.710
Right?

00:09:20.710 --> 00:09:22.360
2 times minus 1 is minus 2.

00:09:22.360 --> 00:09:23.970
1 minus 2 is minus 1.

00:09:23.970 --> 00:09:28.300
And then 2 plus 2 times 9.

00:09:28.300 --> 00:09:29.520
So that's 2 plus 18.

00:09:29.520 --> 00:09:32.190
So that's 20.

00:09:32.190 --> 00:09:35.310
And once again, we could
rewrite this

00:09:35.310 --> 00:09:37.160
equation right here.

00:09:37.160 --> 00:09:39.290
This middle equation right
here is the same thing as

00:09:39.290 --> 00:09:44.000
minus y minus 5z
is equal to 7.

00:09:44.000 --> 00:09:47.070
Just so you remember, this is
just another representation of

00:09:47.070 --> 00:09:48.210
these actual linear equations.

00:09:48.210 --> 00:09:50.890
I just don't feel like writing
the x's, y's, and z's and the

00:09:50.890 --> 00:09:52.750
equal signs.

00:09:52.750 --> 00:09:55.010
But anyway, now let's see
if we can eliminate

00:09:55.010 --> 00:09:56.530
this term down here.

00:09:56.530 --> 00:09:57.810
So how could we do that?

00:09:57.810 --> 00:10:00.970
Well we could add this equation,
we could substitute

00:10:00.970 --> 00:10:05.770
this equation with this equation
plus 6 times this

00:10:05.770 --> 00:10:06.170
one, right?

00:10:06.170 --> 00:10:08.780
Because if you have this plus
6 times this, you'd

00:10:08.780 --> 00:10:10.050
have 6 minus 6.

00:10:10.050 --> 00:10:11.670
And you'd get a 0 here.

00:10:11.670 --> 00:10:14.860
So let's do that.

00:10:14.860 --> 00:10:17.050
Let's write the augmented
matrix.

00:10:17.050 --> 00:10:18.430
We're not changing
the top line.

00:10:18.430 --> 00:10:25.135
So we get minus 1x plus 2y
minus z is equal to 9.

00:10:25.135 --> 00:10:27.690
We're actually not even changing
this equation.

00:10:27.690 --> 00:10:36.210
So we're getting 0x minus y
minus 5z is equal to 7.

00:10:36.210 --> 00:10:40.580
And now we're replacing this
with this plus 6 times this.

00:10:40.580 --> 00:10:44.590
So the 0 plus 6 times 0 is 0.

00:10:44.590 --> 00:10:47.930
6 plus 6 times minus 1.

00:10:47.930 --> 00:10:49.190
That's 6 minus 6.

00:10:49.190 --> 00:10:51.770
That's 0.

00:10:51.770 --> 00:10:57.110
Minus 1 plus 6 times minus 5.

00:10:57.110 --> 00:10:59.220
That's minus 1 minus 30.

00:10:59.220 --> 00:11:02.530
So it's minus 31.

00:11:02.530 --> 00:11:07.510
And then we have 20
plus 6 times 7.

00:11:07.510 --> 00:11:10.200
So that's 6 times 7 is 42.

00:11:10.200 --> 00:11:14.420
So that's 62.

00:11:14.420 --> 00:11:16.640
Now we're actually almost
done solving.

00:11:16.640 --> 00:11:17.390
What can we do here?

00:11:17.390 --> 00:11:20.400
So I just want to make sure you
understand our position.

00:11:20.400 --> 00:11:22.460
What is this last equation?

00:11:22.460 --> 00:11:24.830
This 0, 0, minus 31, line 62?

00:11:24.830 --> 00:11:31.170
Well, this last equation is just
minus 31z is equal to 62.

00:11:31.170 --> 00:11:34.230
And so we can divide both sides
of this equation by

00:11:34.230 --> 00:11:37.850
minus 31, and we'd get z
is equal to minus 2.

00:11:37.850 --> 00:11:40.680
But if we were to do that using
our augmented matrix

00:11:40.680 --> 00:11:43.540
notation, we would just say,
well let's just divide both

00:11:43.540 --> 00:11:47.050
sides of this bottom equation
by minus 31.

00:11:47.050 --> 00:11:56.120
And we would get minus
1, 2, minus 1, 9, 0,

00:11:56.120 --> 00:11:59.810
minus 1, minus 5, 7.

00:11:59.810 --> 00:12:02.640
So we're dividing this
one by minus 31.

00:12:02.640 --> 00:12:08.036
So it's 0, 0, 1, minus 2.

00:12:08.036 --> 00:12:09.560
And there you have it.

00:12:09.560 --> 00:12:12.450
So we have at least
solved for z.

00:12:12.450 --> 00:12:12.810
Right?

00:12:12.810 --> 00:12:15.910
This equation is the same thing
as this one. z is equal

00:12:15.910 --> 00:12:17.570
to minus 2.

00:12:17.570 --> 00:12:19.760
Now at this point, what we
have just done using this

00:12:19.760 --> 00:12:22.490
augmented matrix, this is called
Gaussian elimination.

00:12:22.490 --> 00:12:24.260
Now we can continue this.

00:12:24.260 --> 00:12:26.650
And we can eliminate
this term, this

00:12:26.650 --> 00:12:27.090
term, and this term.

00:12:27.090 --> 00:12:29.600
And it would be called
Gauss-Jordan elimination.

00:12:29.600 --> 00:12:32.690
Or, we could use the fact that
z is equal to minus 2.

00:12:32.690 --> 00:12:35.330
And then we could substitute
back into this equation to

00:12:35.330 --> 00:12:36.480
solve for y.

00:12:36.480 --> 00:12:38.900
And then we'd have our
values for y and z.

00:12:38.900 --> 00:12:40.600
And we could substitute back
into this equation.

00:12:40.600 --> 00:12:43.960
Let's do it that way,
just for kicks.

00:12:43.960 --> 00:12:47.830
So this is called upper
row echelon form.

00:12:47.830 --> 00:12:49.970
I always forget the terminology,
but let's keep

00:12:49.970 --> 00:12:51.010
this equation in mind.

00:12:51.010 --> 00:12:52.610
And let me erase all of
this, so we have some

00:12:52.610 --> 00:12:55.350
room to work with.

00:12:59.110 --> 00:13:02.930
I can't do anything without
some room to work with.

00:13:02.930 --> 00:13:05.685
Let me even erase
this up here.

00:13:05.685 --> 00:13:07.310
And as you can tell,
this is very

00:13:07.310 --> 00:13:08.580
computationally intensive.

00:13:08.580 --> 00:13:11.650
And it's not my cup of tea.

00:13:11.650 --> 00:13:13.375
It's not what I would
want to do if I had

00:13:13.375 --> 00:13:15.880
the weekend to myself.

00:13:15.880 --> 00:13:19.950
But anyway, let me write these
equations in kind of this

00:13:19.950 --> 00:13:23.140
format, so we can kind of go
back to our traditional world.

00:13:23.140 --> 00:13:24.640
So this equation right
here, this is

00:13:24.640 --> 00:13:25.860
actually the same equation.

00:13:25.860 --> 00:13:27.040
We never changed it.

00:13:27.040 --> 00:13:32.890
Minus x plus 2y minus
z is equal to 9.

00:13:32.890 --> 00:13:40.440
This equation is minus y
minus 5z is equal to 7.

00:13:40.440 --> 00:13:45.600
And then this equation is
z is equal to minus 2.

00:13:45.600 --> 00:13:47.270
So we solved for z.

00:13:47.270 --> 00:13:49.250
This equation essentially
solves for z.

00:13:49.250 --> 00:13:51.250
And now we can substitute
back into this one.

00:13:51.250 --> 00:13:59.350
So we have minus y minus 5 times
z, which is minus 2, is

00:13:59.350 --> 00:14:00.840
equal to 7.

00:14:00.840 --> 00:14:05.880
So minus y plus 10
is equal to 7.

00:14:05.880 --> 00:14:09.290
Minus y is equal to minus 3.

00:14:09.290 --> 00:14:11.810
y is equal to 3.

00:14:11.810 --> 00:14:12.420
There we have it.

00:14:12.420 --> 00:14:15.290
We solved for z and y.

00:14:15.290 --> 00:14:19.120
And now we could substitute
the y, the z into this top

00:14:19.120 --> 00:14:20.630
equation, and solve for x.

00:14:20.630 --> 00:14:21.400
So what do we have?

00:14:21.400 --> 00:14:22.730
And then we're done.

00:14:22.730 --> 00:14:23.460
Let's see.

00:14:23.460 --> 00:14:32.310
Minus x plus 2 times y, so 2
times 3, minus z-- minus minus

00:14:32.310 --> 00:14:35.690
2, that's z-- is equal to 9.

00:14:35.690 --> 00:14:44.190
So minus x plus 6 plus
4 is equal to 9.

00:14:44.190 --> 00:14:48.010
Mine x plus 10 is equal to 9.

00:14:48.010 --> 00:14:52.630
Minus x is equal to minus
1, or x is equal to 1.

00:14:52.630 --> 00:14:54.080
So the solution-- oh my God.

00:14:54.080 --> 00:14:55.230
My phone is ringing.

00:14:55.230 --> 00:14:57.190
So many people calling me.

00:14:57.190 --> 00:14:58.140
I think it's my wife.

00:14:58.140 --> 00:14:58.980
She might be downstairs.

00:14:58.980 --> 00:15:00.140
We're supposed to go eat out.

00:15:00.140 --> 00:15:01.140
But anyway, let me
just finish this.

00:15:01.140 --> 00:15:06.080
So the solution to this 3 by 3
equation is the point x is

00:15:06.080 --> 00:15:11.400
equal to 1, y is equal to 3,
and z is equal to minus 2.

00:15:11.400 --> 00:15:15.750
And that's the unique point in
three-dimensional space that

00:15:15.750 --> 00:15:18.010
lies on all three
of these planes.

00:15:18.010 --> 00:15:21.410
Or the unique point where these
three planes intersect.

00:15:21.410 --> 00:15:22.720
Anyway, I hope you got that.

00:15:22.720 --> 00:15:24.420
And it's time for me to go
have dinner with my wife.

00:15:24.420 --> 00:15:25.980
See you soon.


Epsilon-delta limit definition 2

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.680
It is not unusual in calculus
textbooks to see something-- to

00:00:04.680 --> 00:00:11.780
see problems along the lines
of, prove that the limit as x

00:00:11.780 --> 00:00:15.380
approaches-- I don't know,
say, as x approaches 1.

00:00:15.380 --> 00:00:19.240
Let me think of an interesting
function when x approaches 1.

00:00:19.240 --> 00:00:19.380
Let's see.

00:00:19.380 --> 00:00:27.770
3x times x minus 1
over x minus 1.

00:00:27.770 --> 00:00:29.980
And you might say, hey Sal, why
did you do this x minus 1 in

00:00:29.980 --> 00:00:30.820
the numerator and denominator?

00:00:30.820 --> 00:00:32.970
Well, I did that because
obviously you can't

00:00:32.970 --> 00:00:34.020
have a 0 down here.

00:00:34.020 --> 00:00:36.270
So this function is actually
not defined as x is

00:00:36.270 --> 00:00:37.030
equal to 1, right?

00:00:37.030 --> 00:00:39.930
If I had just put a 3x,
this cancels out for

00:00:39.930 --> 00:00:41.120
all the other values.

00:00:41.120 --> 00:00:44.250
So this is essentially the
line 3x, except it has a

00:00:44.250 --> 00:00:45.970
hole at x is equal to 1.

00:00:45.970 --> 00:00:46.580
And that's why I did it.

00:00:46.580 --> 00:00:48.230
Because I want to take the
limit, and it's interesting to

00:00:48.230 --> 00:00:51.070
take a limit a to a point where
the function doesn't exist.

00:00:51.070 --> 00:00:54.200
But the books will say, hey,
prove that this limit is equal

00:00:54.200 --> 00:00:58.650
to-- well, it would actually
be equal to 3, right?

00:00:58.650 --> 00:00:59.440
Prove that.

00:00:59.440 --> 00:01:02.290
And you essentially have
to use the epsilon delta

00:01:02.290 --> 00:01:03.910
definition of a limit.

00:01:03.910 --> 00:01:06.390
So to prove this, you're
essentially proving

00:01:06.390 --> 00:01:07.540
what this means.

00:01:07.540 --> 00:01:11.720
And this means-- proving
this is the same thing as

00:01:11.720 --> 00:01:17.690
proving-- so you could
just prove that given--.

00:01:17.690 --> 00:01:22.190
So if you give me some epsilon
greater than 0-- and remember,

00:01:22.190 --> 00:01:25.980
that's how close you want f
of x to get to the limit.

00:01:25.980 --> 00:01:28.880
Given some epsilon greater than
0, in order for this to be

00:01:28.880 --> 00:01:32.970
true, I need to prove that I
can give you-- that there

00:01:32.970 --> 00:01:38.650
exists-- that, you know, I can
give you a delta

00:01:38.650 --> 00:01:40.400
greater than 0.

00:01:40.400 --> 00:01:49.006
Where as long as the distance
between x and our limit point,

00:01:49.006 --> 00:01:54.090
x and 1, is less than delta--
remember, we don't want x to

00:01:54.090 --> 00:01:56.320
ever get right on top
of the limit point.

00:01:56.320 --> 00:01:58.480
Because a function might
not be defined there.

00:01:58.480 --> 00:02:01.020
So the distance between
x and 1 has to be

00:02:01.020 --> 00:02:02.970
greater than 0, right?

00:02:02.970 --> 00:02:04.390
That's what this tells us.

00:02:04.390 --> 00:02:08.110
So I have to prove, you
give me an epsilon, that

00:02:08.110 --> 00:02:09.850
I can give you a delta.

00:02:09.850 --> 00:02:13.460
And as long as x is within
delta of 1-- and this is all

00:02:13.460 --> 00:02:14.980
this is saying-- as long as
the distance between

00:02:14.980 --> 00:02:15.780
x and 1, right?

00:02:15.780 --> 00:02:17.320
That's when you subtract
2 things and take

00:02:17.320 --> 00:02:17.910
the absolute value.

00:02:17.910 --> 00:02:19.070
That's just distance.

00:02:19.070 --> 00:02:22.050
So as long as the distance
between x and 1 is less than

00:02:22.050 --> 00:02:29.770
delta then the distance between
3x-- let me write that up

00:02:29.770 --> 00:02:37.650
there-- the distance between 3x
times x minus 1 over x minus 1,

00:02:37.650 --> 00:02:39.470
that's the function, right?

00:02:39.470 --> 00:02:44.550
The distance between that and
the limit point-- and 3--

00:02:44.550 --> 00:02:46.260
is going to be less
than the epsilon.

00:02:49.110 --> 00:02:51.390
A lot of calculus text
books, you know, they'll

00:02:51.390 --> 00:02:52.070
write it generally.

00:02:52.070 --> 00:02:57.620
They're like, the limit of f of
x as x approaches a of equals

00:02:57.620 --> 00:03:00.030
l, and so this would be a.

00:03:00.030 --> 00:03:02.410
This would be your f of
x, and this would be

00:03:02.410 --> 00:03:04.090
your l, right there.

00:03:04.090 --> 00:03:06.420
But anyway, when someone says
they want you to prove this,

00:03:06.420 --> 00:03:08.020
they essentially want
you to prove this.

00:03:08.020 --> 00:03:09.410
I'll give you an epsilon.

00:03:09.410 --> 00:03:13.090
You have to prove that any
epsilon that you're given, or

00:03:13.090 --> 00:03:15.860
that you give me, that I--
let's say I'm the one who has

00:03:15.860 --> 00:03:18.360
to prove it-- any epsilon you
give me, I have to be able to

00:03:18.360 --> 00:03:23.380
give you a delta where, as long
as x is within delta of 1,

00:03:23.380 --> 00:03:29.000
then the function is within
epsilon of the limit point.

00:03:29.000 --> 00:03:30.550
And how do we do that?

00:03:30.550 --> 00:03:33.140
And I'll just say, you know, a
lot of proofs, it's just kind

00:03:33.140 --> 00:03:36.600
of-- there's not a-- well,
there's not a systematic

00:03:36.600 --> 00:03:37.250
way to do proofs.

00:03:37.250 --> 00:03:38.520
And that's kind of what
makes them so interesting

00:03:38.520 --> 00:03:39.730
on a lot of levels.

00:03:39.730 --> 00:03:42.060
But these, there kind of tends
to be a systematic way.

00:03:42.060 --> 00:03:45.355
And this problem I've given
you, to some degree, is kind of

00:03:45.355 --> 00:03:48.310
the easy one that they'll give
when they ask you

00:03:48.310 --> 00:03:49.170
to prove these.

00:03:49.170 --> 00:03:51.450
But they usually involve
starting from this point right

00:03:51.450 --> 00:03:55.580
here and algebraically
manipulating it until you get--

00:03:55.580 --> 00:03:57.680
until this expression
right here looks something

00:03:57.680 --> 00:03:59.210
like this expression.

00:03:59.210 --> 00:04:01.640
And then you'll have
this expression is less

00:04:01.640 --> 00:04:04.560
than, you know, epsilon
divided by something.

00:04:04.560 --> 00:04:06.450
Or, you know, some
function of epsilon.

00:04:06.450 --> 00:04:10.450
And it's like, oh, well as long
as delta is that, you give me

00:04:10.450 --> 00:04:13.270
an epsilon, I can just apply
that, you know, take epsilon

00:04:13.270 --> 00:04:14.660
and divide it by
4, or whatever.

00:04:14.660 --> 00:04:16.150
And use that as my delta.

00:04:16.150 --> 00:04:17.710
Then I've proved the theorem.

00:04:17.710 --> 00:04:18.680
And so we'll do that here.

00:04:18.680 --> 00:04:20.510
So let's start with this.

00:04:20.510 --> 00:04:23.810
Let's start with kind of
where we want to get to.

00:04:23.810 --> 00:04:26.380
And then figure out a delta
that's essentially a

00:04:26.380 --> 00:04:27.290
function of epsilon.

00:04:27.290 --> 00:04:30.440
So that whenever I'm given an
epsilon, I can say, hey,

00:04:30.440 --> 00:04:32.380
whatever number you give me,
I'm going to give you back

00:04:32.380 --> 00:04:35.500
a number that's, you know,
that divided by whatever.

00:04:35.500 --> 00:04:37.060
And then this will all work.

00:04:37.060 --> 00:04:37.840
So let's do that.

00:04:37.840 --> 00:04:40.740
So immediately we
can simplify this.

00:04:40.740 --> 00:04:43.750
We can cross these out.

00:04:43.750 --> 00:04:45.720
And when you cross that out,
you still have to say, well, we

00:04:45.720 --> 00:04:49.430
can't do that as long as x--
you can only do this as long as

00:04:49.430 --> 00:04:51.200
you assume that x
does not equal 1.

00:04:51.200 --> 00:04:54.340
Because if x equals 1, this
function is undefined, right?

00:04:54.340 --> 00:04:56.860
And that's OK because we only
care about the intervals

00:04:56.860 --> 00:04:58.610
as x approaches 1.

00:04:58.610 --> 00:05:01.270
We only care about the
intervals as x approaches 1.

00:05:01.270 --> 00:05:03.860
We don't care exactly
when x is equal to 1.

00:05:03.860 --> 00:05:05.660
So that's fine.

00:05:05.660 --> 00:05:10.480
So then this simplifies to--
this right there simplifies

00:05:10.480 --> 00:05:18.550
to-- the absolute value of 3x
minus 3 is less than epsilon.

00:05:18.550 --> 00:05:21.360
And then we could factor out a
3, so that's, you know, we

00:05:21.360 --> 00:05:25.820
could say the absolute value of
3 times x minus 1 is

00:05:25.820 --> 00:05:26.950
less than epsilon.

00:05:26.950 --> 00:05:29.280
And you might already see, we
already have an x minus 1

00:05:29.280 --> 00:05:30.200
there and an x minus 1 there.

00:05:30.200 --> 00:05:33.000
If we can get this 3 over
on this side, then we'll

00:05:33.000 --> 00:05:35.030
be done, essentially.

00:05:35.030 --> 00:05:38.940
So the absolute value of 3
times some other number, it's

00:05:38.940 --> 00:05:43.200
going to be the same thing as
the absolute value of 3 times

00:05:43.200 --> 00:05:46.240
the absolute value
of x minus 1.

00:05:46.240 --> 00:05:48.000
And if you don't believe me,
I mean, try it out with

00:05:48.000 --> 00:05:48.980
a bunch of numbers.

00:05:48.980 --> 00:05:50.600
This is going to be
positive no matter what.

00:05:50.600 --> 00:05:52.040
This is going to be
positive no matter what.

00:05:52.040 --> 00:05:53.850
It's going to have the
same magnitude, right?

00:05:53.850 --> 00:05:56.210
And of course, that is
less than epsilon.

00:05:56.210 --> 00:05:58.600
And this is what we say-- you
know, a lot of times a calculus

00:05:58.600 --> 00:06:01.950
teacher will say, if this is
true, this is true, if and

00:06:01.950 --> 00:06:03.370
only if this is true.

00:06:03.370 --> 00:06:04.830
And that's just a
fancy way of saying.

00:06:04.830 --> 00:06:06.940
So sometimes they'll
do a two way arrow.

00:06:06.940 --> 00:06:11.550
Sometimes they'll write IF
and only IF, kind of an IFF.

00:06:11.550 --> 00:06:13.260
An IF with 2 f's.

00:06:13.260 --> 00:06:14.940
But that just means if
that's true, that's true.

00:06:14.940 --> 00:06:16.570
And if this is true,
then that's true.

00:06:16.570 --> 00:06:18.380
And that's generally the case,
whenever you just do some

00:06:18.380 --> 00:06:19.510
algebraic manipulation.

00:06:19.510 --> 00:06:23.450
You can manipulate this back to
get that, or that to get that.

00:06:23.450 --> 00:06:25.770
And then they'll say, well, if
this is true, this is true

00:06:25.770 --> 00:06:27.370
if and only if this is true.

00:06:27.370 --> 00:06:29.020
But this is just
algebraic manipulation.

00:06:29.020 --> 00:06:29.220
Right?

00:06:29.220 --> 00:06:31.490
Because you can go back and
forth between these steps.

00:06:31.490 --> 00:06:34.720
And that's true, if and
only if that is true.

00:06:34.720 --> 00:06:36.980
And then the absolute value
of 3 is just 3, right?

00:06:36.980 --> 00:06:37.980
We know that.

00:06:37.980 --> 00:06:42.180
Let's divide both sides by 3
and you get-- well, I can

00:06:42.180 --> 00:06:43.920
just get rid of the absolute
value signs right there.

00:06:43.920 --> 00:06:44.725
Because we know that's 3.

00:06:44.725 --> 00:06:46.760
If you divide both sides of the
equation by 3, you get the

00:06:46.760 --> 00:06:51.680
absolute value of x minus 1
is less than epsilon over 3.

00:06:51.680 --> 00:06:53.120
So what have we
just done so far?

00:06:53.120 --> 00:06:59.455
We've just proved that the
absolute value-- I need to

00:06:59.455 --> 00:07:00.710
write this in another color.

00:07:00.710 --> 00:07:04.930
We've just proved that the
absolute value of essentially

00:07:04.930 --> 00:07:14.270
3x times x minus 1 over x minus
1 minus 3 is less than epsilon,

00:07:14.270 --> 00:07:22.620
if and only if x minus 1 is
less than epsilon over 3.

00:07:22.620 --> 00:07:23.390
Right?

00:07:23.390 --> 00:07:29.570
So it's like, this is true, as
long as the distance between x

00:07:29.570 --> 00:07:32.890
and 1 is less than
epsilon over 3.

00:07:32.890 --> 00:07:36.080
So we can just use epsilon
over 3 as our delta.

00:07:36.080 --> 00:07:39.100
Remember the whole point of
this was like, you give me

00:07:39.100 --> 00:07:42.645
an epsilon, you give me a
distance, or you say, I want

00:07:42.645 --> 00:07:46.110
to be within this distance
of my limit point.

00:07:46.110 --> 00:07:48.480
I want the function to be that
close to the limit point.

00:07:48.480 --> 00:07:51.100
And this is just the distance
between the function and

00:07:51.100 --> 00:07:52.240
the limit point, right?

00:07:52.240 --> 00:07:55.150
This is f of x.

00:07:55.150 --> 00:07:58.150
And this is the limit
that it approaches.

00:07:58.150 --> 00:07:59.360
So you want to be that close.

00:07:59.360 --> 00:08:02.680
And I say, well, you're going
to be that close if and only if

00:08:02.680 --> 00:08:06.600
x-- the distance between x and
1, or the distance between x

00:08:06.600 --> 00:08:08.320
and the value it's
approaching-- is

00:08:08.320 --> 00:08:08.750
less than this.

00:08:08.750 --> 00:08:10.630
So we just prove
this algebraically.

00:08:10.630 --> 00:08:13.890
So whatever number you give
me-- if you give me a-- if

00:08:13.890 --> 00:08:16.490
you say, Sal, I want to be
within 1 of the limit point.

00:08:16.490 --> 00:08:20.110
I want f of x to be no further
than 1 from the limit point,

00:08:20.110 --> 00:08:22.850
then I'll say, OK, and
I'll give you 1/3, right?

00:08:22.850 --> 00:08:24.780
Because this is true.

00:08:24.780 --> 00:08:26.580
Let me write that out.

00:08:26.580 --> 00:08:30.460
So let's say you pick
1 as an epsilon.

00:08:30.460 --> 00:08:37.597
So you say, hey, you need to
prove to me that there's some x

00:08:37.597 --> 00:08:43.080
value where, as long as the x
value is no more-- there's

00:08:43.080 --> 00:08:44.130
some value of delta.

00:08:44.130 --> 00:08:49.080
So as long as we're no more
than delta away from 1, then

00:08:49.080 --> 00:08:52.210
the function itself will
be no more than 1 away

00:08:52.210 --> 00:08:53.570
from the limit value.

00:08:53.570 --> 00:08:54.590
Right?

00:08:54.590 --> 00:08:56.660
And then I say, well,
you, gave me an epsilon

00:08:56.660 --> 00:08:58.310
that's equal to 1.

00:08:58.310 --> 00:09:01.140
So I could just make-- I can
just say this is, you know, we

00:09:01.140 --> 00:09:04.860
just proved, if and only if
absolute value of x minus 1 is

00:09:04.860 --> 00:09:09.120
less than whatever number
you gave me divided by 3.

00:09:09.120 --> 00:09:10.790
And that actually makes
a lot of sense if you

00:09:10.790 --> 00:09:13.900
graph this equation.

00:09:13.900 --> 00:09:14.990
Let me see.

00:09:14.990 --> 00:09:20.100
Let me draw the x-axis, y-axis.

00:09:20.100 --> 00:09:20.710
We already established.

00:09:20.710 --> 00:09:24.470
This thing looks just like the
equation of 3x except it has

00:09:24.470 --> 00:09:27.490
a hole at x is equal to 1.

00:09:27.490 --> 00:09:29.840
Let me draw the graph.

00:09:29.840 --> 00:09:31.530
It'll have slope.

00:09:31.530 --> 00:09:32.700
It'll have the slope of 3.

00:09:32.700 --> 00:09:33.790
So it'll be pretty steep.

00:09:33.790 --> 00:09:36.410
It'll look something like that.

00:09:36.410 --> 00:09:40.940
And when we're at 1--
this point right

00:09:40.940 --> 00:09:43.200
here-- there's a hole.

00:09:43.200 --> 00:09:43.420
Right?

00:09:43.420 --> 00:09:45.620
This is 3.

00:09:45.620 --> 00:09:49.880
So all we're saying, we just
proved that-- so if you said

00:09:49.880 --> 00:09:52.930
that you wanted to be within 1
of our limit point, that's like

00:09:52.930 --> 00:09:55.970
you giving me an epsilon of 1.

00:09:55.970 --> 00:09:58.710
So this distance
right here is 1.

00:10:02.090 --> 00:10:03.780
Which would be a
pretty big number.

00:10:03.780 --> 00:10:06.970
So you want the function
to be within 1 of our

00:10:06.970 --> 00:10:07.980
limit point, right?

00:10:07.980 --> 00:10:10.240
Because this is
the limit point.

00:10:10.240 --> 00:10:13.510
Then all we're saying is, you
just have to take 1/3 of that.

00:10:13.510 --> 00:10:16.520
So we just have to be within
1/3-- so this distance right

00:10:16.520 --> 00:10:22.570
here is 1/3 away from 1.

00:10:22.570 --> 00:10:25.030
Or we could have said it more
generally, this is epsilon.

00:10:25.030 --> 00:10:26.180
This is epsilon.

00:10:26.180 --> 00:10:28.220
This would be epsilon over 3.

00:10:28.220 --> 00:10:29.810
This is epsilon over 3.

00:10:29.810 --> 00:10:32.090
And the reason why this is a
proof, is because it shows no

00:10:32.090 --> 00:10:35.670
matter what epsilon you give
me, I can always find a delta.

00:10:35.670 --> 00:10:37.410
Because I can just take
whatever number you gave

00:10:37.410 --> 00:10:38.410
me, and divide it by 3.

00:10:38.410 --> 00:10:41.160
And I can divide any real
number greater than 0.

00:10:41.160 --> 00:10:42.690
I can divide any
real number by 3.

00:10:42.690 --> 00:10:44.970
So I can always-- no matter
what you give me-- I can always

00:10:44.970 --> 00:10:45.840
find you something else.

00:10:45.840 --> 00:10:49.640
And therefore, I have proven by
the epsilon delta definition of

00:10:49.640 --> 00:10:52.350
limits that the limit
as x approaches 1, of

00:10:52.350 --> 00:10:54.890
this, is equal to 3.