This brain teaser in this video is once again not one
that I made up.
I actually don't remember the first time that I heard it.
I think it was in the back of a magazine.
I want to say Scientific American.
And me and a bunch of buddies, we were
traveling through Europe.
And it kind of became our pastime to do these brain
teasers in the back of magazines.
And just so you know, I think a lot of you think that these
type of brain teasers, you can just read them and solve them.
And if you can't solve them, then you're not good at this
type of thing.
This brain teaser I'm about to give you, I just kind of sat
and thought about it, and me and my buddies, we argued
about it for literally about a day.
And then I just slept on it.
And then first thing in the morning, I woke up all my
friends, I was like, I figured it out!
So this isn't some kind of easy thing.
And I encourage you, just listen to the statement,
understand it, and then pause the video.
And then think about it for 24 hours.
And then come back to it.
One, you'll probably solve it yourself.
And if you don't, in 24 or 48 hours, then you'll probably
find the solution that much more satisfying.
So this is the problem of the togglers.
Not toddlers.
Togglers.
I have an infant, so words like toddler are in my brain.
But this is the problem of the togglers.
So there are five guys in front of you.
That's one of them.
Two.
Three.
Four.
Five.
They don't have to be guys.
They can be of any gender.
And four of them are what we call togglers.
So four of them.
Four togglers.
And what a toggler does is, the first time you ask them a
question, they're either going to tell the truth or they're
going to lie.
So truth or a lie.
But then the second time that you ask them,
they're going to toggle.
They're going to switch.
So the second time you ask them a question, if they told
the truth the first time, then the next time you ask that
same person a question, they're going
to switch to lying.
But if they lied the first time you asked them a
question, then they're going to switch
to telling the truth.
And then it goes on, and so on, and so forth.
So if you ask the same person the third question, if they
told the truth the first time, then they'll lie the second
time you ask them a question, and then the third time
they'll tell the truth again.
It's like, if I was a toggler and you said, hey you,
what is your name?
And if I said, oh my name is Bill.
Then clearly I'm lying that first time.
And then you pointed to me again, and you say,
what is your name?
And I say, oh my name is Sal.
And if you ask me again, what is your name?
I would say, oh, my name is Susan.
And if you ask me again, my name is Sal.
So the first time, I just decided to lie.
But just as easily, I could've decided to tell the truth the
first time.
You might have said, hey you, what's your name?
I would have said, Sal.
And then you ask me, what's your name again?
And I would have said Elizabeth.
And then you ask me, what's my name again?
I would say, oh, my name is Sal.
And then again, what's your name?
And I would say, you know, George Washington
or something arbitrary.
But I think you get the idea.
A toggler-- I don't know who's a toggler.
Let's say that this is a toggler.
The first time you ask them a question, they might lie, they
might tell the truth.
But if they lie, the next time you ask the same person a
question, they're going to tell the truth.
But the first time you ask them a question, and they did
tell the truth, the next time you ask the same person a
question, he or she is going to lie.
So that's a toggler.
So you can already imagine, this is-- I don't know.
The first time that I read the problem it kind of made my
brain hurt.
This notion of a toggler.
It's much easier when there's a reality where people only
lie or only tell the truth.
It's much harder when they toggle.
And not only do they toggle, but their initial state,
whether they start telling the truth or they start off lying.
You don't know.
That's arbitrary.
There's no way of predicting whether this person-- First of
all, we don't know if this person is a toggler, because I
said there's only four togglers.
But even if they are toggler, I don't even know if the first
time I ask them a question whether they're going to tell
the truth or whether they're going to lie.
So four of them are togglers.
What's the fifth?
Well, you probably could guess because this is how these
brain teasers go.
The fifth is a truth teller.
And he is much simpler.
Or he or she.
I don't want to give any clues on who it might be.
Is a truth teller.
And a truth teller always tells the truth.
So the goal of this brain teaser is to figure out who is
the truth teller.
So you want to know which of these five people is-- So
let's say, goal.
Who's truth teller?
And as you can imagine, you don't have infinite questions
to do this with.
You have exactly two questions to do this with.
And you have two questions.
And those two questions, they can be one question to this
guy and another question to this guy.
Both questions could be to this guy.
Both questions could be to her right over there.
So the questions don't have to be to the same person,
although they can be to the same person.
Or they could be to two different people.
So you have two questions.
Any question to ask of any combination of either two
questions to one person or two questions to two people.
And by the end of those two questions, you should know for
sure who is the truth teller.
And that is the brain teaser statement.
So pause it now if you don't want any hints or the
solution, or anything else.
So I encourage you to pause it and think about it for at
least a day.
And sleep on it.
Because a lot of the real math problems or logic problems
that are worth solving aren't the ones that you can solve
right when you see them.
They're the ones where you kind of sit and let your brain
do a little background processing while you sleep.
So that said, hopefully you've stopped it and now you've
resumed it after really struggling with this for a
long period.
Not eating and sleeping and all the things that
normal people do.
And now you're ready for the solution.
Or even better, you're ready to confirm your solution.
So first I'll give you a hint.
I mean, it's a tough question.
I struggled with this at first. I was like, when you
ask someone a question how do you know if
they're lying or not?
How do you know if they're the truth teller?
How do you know if they're a toggler?
You don't have any information.
So the crux of this problem is, essentially, it doesn't
matter who you're asking the question to
the first time around.
And actually the solution is that it doesn't matter who
you're asking the question to the second time around.
You want to ask a question so that no matter what they say,
you know what they're going to do the second time around.
You know whether they're going to tell the truth the second
time around, or you know whether they're going to lie
the second time around.
And that's a hint.
And if you just want a hint and then to keep thinking
about it, pause it again.
Because I'm about to give you the solution.
So question one.
And you can pick any of the five people to ask this
question to.
And that's kind of the nature of one of these solutions.
There's no reason for you to pick one of the
five over the other.
They're all equally likely to be the truth teller.
So you ask them, are you the truth teller?
So let's think about a couple of-- Well first of all,
they're either going to answer to this, yes or no.
So there could be two situations.
They're either going to say yes, or they're
going to say no.
So if someone says yes to are they the truth teller, it
means that either they are the truth
teller telling the truth.
Or they are a lying toggler.
If they say that they are not the truth teller, they clearly
cannot be the truth teller, because the truth teller
cannot lie about not being the truth teller.
And so they have to be a toggler.
And they'd be an honest toggler.
They'd be a toggler who's telling the truth.
Right?
So it would be a truthful toggler.
Remember, I ask anybody, any of these five people right
here, say, are you the truth teller?
If they say yes, I immediately know that they're either the
truth teller telling the truth, or
they're a lying toggler.
If they say no, I know that they are a truthful toggler.
Now, what's interesting is, in this scenario, if they say
yes, what do I know that they're going to do next?
So if this is the truth teller, and they told the
truth, on the second question, they're going to tell the
truth again.
They're going to tell the truth on question number two.
If they're the lying toggler, what are they going to do when
you ask them another question?
Well, they toggle.
If they lie the first time you ask them, they're going to
tell the truth the second time you ask them.
So they're going to tell the truth on question number two.
So if the person says yes to, are you the truth teller?
You immediately know that no matter who they are, that
they're going to tell the truth on question number two.
So if you can know for a fact that the person's going to
tell the truth on question number two, you ask them, who
is the truth teller?
So this is number two.
Who is the truth teller?
So it's kind of weird.
You go to someone and say, are you the truth teller?
And they'll say yes.
And then you ask them again, who's the truth teller?
And then, because they have to tell the truth, they will
actually either say, if it was this case, they'd say I am the
truth teller.
Or if it's this case, they'd say no, that dude over there
is the truth teller.
And then you'll know who the truth teller is.
Now what happens in this situation?
That they're a truthful toggler?
Well on the second question, what are they going to do?
They're going to lie.
They're going to lie on the second question, right?
On question number two.
So if you know for a fact that someone's going to lie, how
can you phrase a question so you can figure out who the
truth teller is?
Well, if they say no to this first question, and you know
they're going to lie on the second one, your second
question should just be, who is not the truth teller?
So the only way to lie on this question-- and this person has
to lie now-- is to tell you who the truth teller is.
And so, using the first question, are
you the truth teller?
We can immediately predict what the person's going to do
on the second question.
And then we can phrase the second question, depending on
that response, to definitely know who the truth teller is.
Anyway, I thought you'd enjoy that.
And hopefully you see why the liar game show brain teaser
was a bit of a warm up for this one.
See you in the next video.