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- Ten of us have been abducted by aliens because they view us
- as being representative of the human race.
- And what they're going to do is they're going to test us to
- see if the human race is worthy of joining the
- intergalactic council of planets.
- And if we're not, they're going to kill us all and and
- turn Earth into a mushroom farm, because
- that's their oil.
- So they take us together in their spaceship and they say,
- you ten represent all of humanity, and the fate of
- humanity rests in your hands.
- And we'll give you a test. And the test is tomorrow.
- 24 hours from right now, we're going to put you all in a
- completely dark room, and line you all up
- one behind the other.
- So if this is one guy-- let me see if I can draw-- so if this
- is the guy in the front.
- He's looking a little worried.
- Someone described me as the Bob Ross of mathematics.
- So I'll attempt to do a little Bob Ross work right here.
- So this would be the first of us.
- He has a business tie on, and whatever.
- He's facing in this direction.
- And then there's ten more of them.
- So instead of redrawing them, let me just copy and paste it.
- We all look mysteriously similar to each other.
- That's the second one.
- And the third one.
- There's ten of us like this.
- I'll do one more.
- Oh that's good enough.
- There's ten of us, and we're essentially lined up front to
- back, so that this guy, all he sees is the back
- of this guy's head.
- And he can also see the back of this guy's head.
- Maybe there's a slight incline, maybe the
- floor is like this.
- Let me see if I can draw it.
- The floor is at a slight tilt, so one guy is
- standing right here.
- Another guy is standing right here.
- Another guy's standing right here.
- There's ten of us in a line.
- We're all facing in that direction.
- All this guy right here can see is the wall.
- This guy can see the two guys in front of him.
- This guy can see the one guy in front of him.
- There's ten of them.
- It goes all the way back to ten
- So what they tell us is we're going to
- line you up like this.
- They even draw a nice diagram like this, because they want
- to give us a fair chance of preventing Earth from turning
- into a mushroom farm.
- And they say, while the room is completely pitch dark,
- we're going to place a hat on each of your heads.
- And the hat will either be mauve-- so this
- would be a mauve hat.
- Or they'll be green.
- So they either get a mauve or a green hat on the
- back of your head.
- And the hat will be designed in such a way, it
- won't have a big rim.
- So that you can't look up and see your hat.
- This is alien technology, so it is completely impossible to
- see the hat on your own head.
- And the distribution of the hats will be random.
- It could be all green hats.
- It could be all-- well, I'll say purple-- because I'm not
- quite sure how to pronounce mauve.
- So it could be all purple hats, it could be half purple,
- half green.
- It could be nine green and one purple.
- They don't know.
- It could be any combination of hats.
- And what they're going to do-- and they tell us all of this--
- is that they're going to start with the person in the back.
- So this is the front wall, this is the floor.
- This is the first guy right here, second guy, third guy,
- fourth guy, fifth guy, sixth guy, seven, eight, nine.
- So there's the tenth guy.
- They're all faced in that direction.
- They're going to start with this guy right here.
- And say, what color is the hat on your head?
- And he'll answer, either purple or green.
- He can only say purple or green.
- I have to be exact here.
- He can only say purple or green.
- And he can say it as loud as he wants.
- But he has to say it at an even tone, so that not more
- information can be conveyed, other than the fact that he
- thinks his hat is purple or green.
- Although this guy is close enough that he can hear it.
- Everyone's going to hear what everyone's going to say.
- But there's no intonation.
- And if you're right, if you say, my hat is purple and you
- are wearing a purple hat, you will live.
- And the 600 million people that you
- represent will also live.
- And where they live will not be turned
- into a mushroom farm.
- And if you're wrong, then they will immediately-- what is
- that Star Trek weapon?
- Displace you.
- And they'll also do it to all the people you represent.
- And turn all of their land into mushroom farms. So
- there's this huge incentive to try to get this right.
- So they say, we're going to do this tomorrow.
- And your goal, obviously, is to try to save as many lives
- as you can.
- And so we're going to put you in a room, and you guys can
- discuss this for the next 24 hours, to come up with a
- system so that when we put you in the dark room and we put
- the hats on each of your heads.
- I shouldn't be drawing brown hats.
- We'll put the hats on your head.
- And then turn on the lights.
- I forgot to tell you that.
- They're going to put us in a dark room, you can't see
- anything, we're all lined up.
- Then they're going to turn on the lights.
- And so then they'll ask this guy, what color is
- the hat on your head?
- He obviously can't look at his own head, but he'll see
- everyone else's hats.
- And likewise this guy, he can't see his own hat, and he
- can't see the guy behind him.
- But he'll hear what this guy says, and he can see all of
- the hats in front of him.
- So they'll turn on the lights, and then after this guy-- if
- he's right he lives, if he's wrong he'll be displaced.
- And then they'll go to the next guy.
- And then ask him.
- If he's right he lives, and all the people
- he represents lives.
- If he's wrong he also will get displaced.
- And they'll keep going all the way up to the first guy.
- So our goal to essentially save civilization, or at least
- human civilization, is to come up with a system to save as
- many of us as possible.
- So that's the first take of the brain teaser.
- So if you want to kind of pause it right here.
- And actually before you pause it, let me just clarify.
- All you can say is either purple or green.
- No intonations.
- You can't say "pur-puuul" or "purp-l" to kind of convey any
- type of information.
- All you can say is purple or green.
- Or pronounce it in some weird way.
- And there's no weird trick where you can kind of look
- with the side of your eyes or anything like this.
- You have to do it completely based on what you hear and
- what you say.
- And if the aliens catch any hint of you
- guys trying to cheat.
- Or all of us.
- I'm one of us.
- To cheat this, they'll just evaporate everything and turn
- everything into mushroom farms.
- So that's the problem statement.
- How do you save as many people as possible?
- And if you want to pause now, feel free to do so.
- And now I'll give you kind of a small hint, because this
- doesn't help you.
- It's easy to think of a lot of solutions that might save a
- couple of people.
- But there's actually a solution that will guarantee
- saving nine people.
- So that's the solution that you have to come up with.
- And for the rest of this video I'm just going to keep giving
- hints, and then eventually give you the solution.
- So the goal is actually really to save nine people.
- And you can which nine people it's going to be.
- And so once again, pause it if you don't want any more hints.
- It's going to be these nine people.
- These nine people are guaranteed to be saved.
- And this guy obviously is just going to be guessing.
- He is the first guy to be asked what color his hat is.
- He has no information.
- He can see the people in front of him, but he has absolutely
- no information as to what color this hat is.
- So he just guesses.
- So there's a 50% chance that he's right.
- Without any other information, we assume that there's a 50-50
- chance that the hat on the top of your head is a
- purple or a green one.
- So your expected number of people you're going to save is
- nine and a half.
- With the system.
- So these people are guaranteed to be saved.
- And so, there's a couple of things that are more
- probabilistic that don't quite get you there.
- That might come to mind immediately.
- And once again, stop this video.
- Because it's fun to think of these yourself.
- But this guy might just see, OK how many green hats are
- there, and how many purple hats are there?
- And then whichever there are more of, he might say, oh my
- hat is-- let's say there's six purple and three green in
- front of him-- he might say my hat is purple.
- And he's either right or wrong, and
- he might get displaced.
- But at least he'll have conveyed that information to
- the rest of the group.
- That hey, the majority of the hats are purple, so you're
- better off guessing purple than green, and it'll save at
- least six of these people because
- they'll all guess purple.
- And that's OK, and in this situation you're guaranteed to
- save more than half of these nine people, so you're
- guaranteed to save at least five of these nine people.
- And then probablistically maybe you can get
- up to seven or eight.
- Then there's the other method.
- Well, there's a ton of them.
- I don't go into them because I realize I don't
- have a lot of time.
- So here is the real deal solution.
- So really pause it and wait and watch it tomorrow, unless
- you don't want the solution.
- The guy in the back, what he does.
- So there are a bunch of people.
- One, two, three, four, five, six, seven,
- eight, nine, ten people.
- And they're all wearing hats.
- And remember, we have 24 hours to come up with a system.
- So everyone has the system.
- And we don't know who's going to be in what order also.
- That's the other thing.
- But that actually doesn't help or hurt you much anyway.
- So the guy in the back, what he does is he counts the
- number of green or purple hats.
- Right?
- This is the system we've come up with.
- And so this is true of whoever's in the back.
- Whoever's in the back has to do this, right?
- So if he counts that the number of green-- let's say
- this is our system.
- He counts the green actually only.
- So count the green.
- And he determines that the number of
- green is an odd number.
- If he determines that the number of green that he sees
- in front of him is an odd number, then he is
- going to say green.
- For the guess on the top of his head.
- If the number of green is an even number, he's
- going to say purple.
- And of course, he doesn't know what his hat color is, but at
- least, well, one, he has a 50-50
- chance of saving himself.
- But now, what happens?
- Now, by giving this information to the rest of the
- group, by screaming green or purple.
- If this guy screams, my hat is green!
- Then this guy, who's the next guy in line, the next guy to
- be quizzed, knows that the person behind him saw an odd
- number of green hats.
- Now this is interesting.
- If he counts the number of green in front of him, and he
- also sees an odd number of green hats, then that means
- that he was not one of the green hats that this guy saw.
- So he must be a purple hat.
- So he'll just say, I have a purple hat.
- And he will live.
- Right?
- And then you'll go to this guy.
- And let's go the other situation.
- Let's say that this guy, he screamed green.
- Which means that he saw an odd number of green
- hats in front of him.
- If this guy counts the green hats of the eight people in
- front of him, and he says wow, there's only an even number of
- green hats, he could say, wow I must have a green hat.
- And so then he'll say green.
- And then he'll be right.
- Likewise, this guy-- let's say it's the first situation that
- I said, this guy has a purple hat on.
- So this guy saw an odd number of green hats.
- He screamed green.
- And so this guy, he also sees an odd number of green hats,
- because he's not one of the green hats.
- So he says purple, he lives.
- This guy all of a sudden, because this guy said purple.
- And he lived.
- I mean, you didn't hear any buzzer.
- Actually you know he's going to live, because
- the system will work.
- He knows that there must be an odd number of green hats from
- him down forward, right?
- Because this guy didn't say green.
- So if he knows that there's an odd number of green hats from
- him down forward, and if he sees an even number of green
- hats, he knows that he must be one of the green hats, so
- he'll say green.
- And in that situation, if this guy screamed green then that
- means that if this guy saw an odd number of green hats from
- here on down, and one of these guys behind him, or every time
- someone behind you says green after that, then the number of
- green hats that are odd or even in front of that person
- will change.
- And so they have to keep track of that in their minds to
- figure out if the hats in front of them are going to be
- green or purple.
- Actually let me give an example because that's
- probably the easiest way.
- So there's one, two-- Actually let me just do it in a neutral
- color so I don't waste time.
- So let's say the hats are green, purple, purple, green,
- green, purple, green, green.
- How many is that?
- One, two, three, four, five, six, seven, eight, nine, ten.
- So you go to the first guy right here.
- And so he counts.
- He says, OK, how many greens are there?
- There's one, two, three, four, five.
- There are five greens.
- So the greens are odd.
- Which says that he is going to say green.
- That's the system we've come up with.
- So when they go up to this guy, he says green.
- And he's wrong.
- So he actually gets eliminated.
- And maybe the aliens were willing to give us a pass on
- the first guy, but let's say he gets eliminated.
- But he said green, so then this guy on down knows that in
- front of that guy there was an odd number of green hats.
- Now, this guy, he looks forward and he counts one,
- two, three, four, five green hats.
- So he also sees an odd number of green hats, so he knows
- that he must not have a green hat, because if he had a green
- hat, then it would turn the number of green hats that he
- saw into even.
- So he doesn't have a green hat.
- He says, I have a purple hat.
- And he lives.
- Now this guy, he counts the number of green
- hats in front of him.
- One, two, three, four green hats.
- He says, well gee, I only see an even number of green hats.
- Up until this guy, he saw an odd number of green hats.
- I must be a green hat.
- So he says green.
- Check.
- Now this guy's turn.
- So first of all, he knows how many green hats this guy saw.
- He knows that this guy saw exactly four green hats.
- Because if this guy had seen five green hats, he would have
- said purple.
- So this guy knows that this guy saw four green hats.
- But he sees only one, two, three green hats.
- Right?
- So he knows that he must be one of the green hats.
- So he screams green.
- So check.
- So then this guy knows, wow there must be only three green
- hats from him on down.
- Because this guy must have seen three green hats.
- If he saw four, he would have said purple.
- So that guy uses that same logic to say OK, there must be
- three green hats from me on down.
- I see three green hats, so I must not have one of them.
- I must be purple.
- And you just keep on going down all the way until
- everyone answers correctly.
- And you save at least nine people.
- And the bulk of humanity.
- And if you gave a pass on the first guy,
- you save all of humanity.
- Anyway, I hope you found that one enjoyable.
- This is one of my favorite brain teasers.
- And I apologize because I realize when I was trying to
- explain it without the example, it might have gotten
- a little confusing.
- But hopefully this example explains things
- a little bit better.
- Although it might be confusing to see green and purple
- written in blue.
- See you in the next video.