Toggler Brain Teaser

Separating the truthteller from the togglers!

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.