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# Liar truth-teller brain teaser

How to get correct advice when you can only ask one question from either a liar or a truth-teller (when you don't know who is what). Created by Sal Khan.

Video transcript

This brain teaser is a bit of
a classic, and I'm sure a bunch of you have
heard it before. But I wanted to include this in
the brain teaser playlist that I'm making right now. Because this one is actually a
good build up, and a bit of a nice warm up session for a
slightly harder brain teaser that deals with people telling
the truth and lying. So this one, we could call it
the truth teller and the liar. So we're on some type of
adventure quest, and we get to a fork in a road, right? So let's say that there's
two doors. That's even better. So there's two doors. And I've gotten to the end of my
adventure quest. And I know that my treasure is behind
one of these two doors. Unfortunately, behind the other
door is slow and painful death, and all of the
misery in the world. So I want to pick the
door carefully. And beside the doors, there are
two people sitting there. There's one guy that
looks like this. And then there's another guy
that looks like this. And I know for a fact - the
gods of brain teasers have come and told me - that both of
these guys know which door has the treasure, and both of
these guys know which door has the unbelievable despair
that would ensue if I opened that door. So both of these
guys know which door I should be opening. And I also know, and both of
them know, that one of them always tells the truth. One is always honest. And then
one of them always lies. And the problem is I don't
know who's who. And I only have one chance. Obviously I can't just open
a door and close it. If I open the wrong door,
the despair will come out and kill. And you know, all of the
pain will happen. And I can only ask one question
of one of them to decide which door to open. So remember, they both know
the correct answer. But one of them always lies. One of them always
tells the truth. And I only have one question
to ask of one of them. So the brain teaser is: What
question do you ask? Who do you ask it to? Do you ask it to the green
guy or to the brown guy? And then what do you do based
on what they tell you? So that is the problem
description. And I will now give
you the solution. So if you don't want
the solution, and I encourage you not to. I think if you think about this
one for maybe 20 or 30 minutes, or at most a day. Sometimes when you sleep
on a brain teaser it might come to you. So I encourage you to do that. So pause it, stop
it, whatever. Don't listen anymore. Because here is the solution. So what you do is-- I mean,
you don't know who's who. So it's not like there's a
correct answer on who you ask a question of. But what you do is you ask you
either of the people, what would the other guy say is
the correct door to open? So let's think about this. So I could ask this guy, I'm
asking the green guy, what would this guy say is the
correct door to open? Now what happens if this
guy tells the truth? If this guy tells the
truth, then this guy's the liar, right? And so he will actually
tell me what this guy would tell me to do. So what would the other guy
say is the correct door to open, right? So this guy, if you were to tell
him the correct door to open, he would tell you the
incorrect door to open. Right? So when you ask this guy what
would that guy say, he'll be honest about what this
guy would say. And since he's a liar,
he would tell you the incorrect door. So the answer to this question,
you would get the incorrect door. Incorrect door would be the
answer to that question if this guy's the truth teller
and this guy's the liar. Now what happens if it's
the other way around? What if I ask this guy and he
happens to be the liar? So this guy's the liar, this
guy's the truth teller. So the liar, he can't
help himself. He will lie. So he knows that this other
guy is the truth teller there, right? There's only two
of them there. He knows this guy's
the truth teller. And he knows that this guy
when asked will say the correct door to open. So if you happened to be lucky
and you ask this guy the correct door, he would
tell it to you. This guy will lie. Let's say that this is the
correct door, door number two is the correct door. So let's say that door number
two is the correct door. If you ask this guy directly,
he'll say oh yeah, door number two. You'll find all your happiness
behind door number two. Now if this guy's a liar, he
knows that the truth teller will say door number two. But he's a liar, so he'll tell
you door number one. So once again, if I ask the same
question of the liar, I still get the incorrect door. Right? So no matter who I ask this
question to, their answer will be the wrong door, the incorrect
door, or the door that I get all of
the pain from. So what you do is you ask this
question, what would the other guy say is the correct
door to open? And no matter who you ask that
to, they're going to answer the incorrect door. And so you do the opposite. So I ask this guy, mister green
guy, what would mister brown guy say is the correct
door to open? And green guy will say brown guy
will tell you to open door number one. Well green guy says that brown
guy will tell me to open door number one, then I should
open up door number two. And likewise, if green guy says
brown guy will say to open door number two, then I
should open door number one. Anyway that's a nice warm up
brain teaser for the next one I'm going to give you.