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### Course: Math for fun and glory > Unit 2

Lesson 1: Brain teasers- Cheryl's birthday
- Heavier ball
- Liar truth-teller brain teaser
- Toggler brain teaser
- Alien abduction brain teaser
- Blue forehead room brain teaser
- Blue forehead room solution
- Forehead numbers brain teaser
- Light bulb switching brain teaser
- Path counting brain teaser
- 3D path counting brain teaser

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# Forehead numbers brain teaser

The perfect logicians are at it again. Created by Sal Khan.

## Want to join the conversation?

- Could you solve this brain teaser with more than 3 numbers, and could those numbers be anything other than the ones that were suggested in this video???(166 votes)
- Yes but it would be much harder with even 1 more person.(1 vote)

- I doesn't seem to work out for me if I choose numbers other than 20 and 30 (or another 2/3 ratio. For example with 25 and 45. I can be 20 or 65, so the last guy will be 5 or 65 if I am 20, or 40 or 90 if i am 65. All those options are legal, so I will still be undecided. So I am thinking it only works when b-a = a-(b-a) or when a=2/3b. Am I right here?(15 votes)
- A big part of the problem is that you are able to figure out what your number is knowing the other two can't figure out their number. This won't be the case for any 3 numbers.(4 votes)

- is the answer 50 because it feels like the unknown is the sum of the other numbers(11 votes)
- It does not have to be that way but yes.(1 vote)

- Hey, here's a brain teaser/lateral thinking puzzle of my own: There are 2 rocks. Both of them are EXACTLY 1 billion years old, yet one is older than the other. How?

In a few days, I will be posting the answer, if none of you guys can figure it out.(10 votes)- well, you'd be... sort of right. i mean, if they were both exactly 1 billion years old, maybe one of them was still aging, one second older. because there's no such thing as the present. how? think about it. if you were to stop time, you'd be frozen too. but put it to normal, everything i in the past. when you talk, you say it, and it's in the past.(1 vote)

- Here's a brain teaser-

3 perfect logicians are taking a nap under a tree. While they are napping, a small boy smears their noses with red berries. When they wake up, they begin to laugh, thinking the other two were laughing at each other. But then, one of them stops laughing, realizing that his nose is red too. How did he come to this conclusion?

/!\ WARNING!

The answer is not too obvious.(4 votes)- Let’s call the philosopher’s A, B and C. A reasoned that B was confident his nose wasn’t red. If B saw A’s nose wasn’t red, he would be surprised that C was laughing, because C would have nothing to laugh at. But B wasn’t surprised, therefore, A correctly reasoned his nose was smeared.

XD(3 votes)

- at7:17is minus 10 equal to negative 10? This brain teaser was awesome!(4 votes)
- Yes, subtracting ten is also equal to negative ten, if you've done algebra before.(1 vote)

- A bomb just exploded in the middle of a city. 32 people survived and over 20,000 people died. The survivers needed supplies to live. One brought water, one brought food, one brought clothes, one brought a pet, and one brought a phone. The rest of the people were thinking on what to do. One person said they could go to texas and ask for help. The other option would be to call for help from the guy who brought the phone. Imagine you were in this situation, what would you do given the supplies you have.(3 votes)
- One would assume that this was not New York City. If I drove to Texas from New York City, it would be a long time.(2 votes)

- What do u mean by forehead numbers(3 votes)
- The contestants of this game have numbers painted on their foreheads.(1 vote)

- if i see 20 or 30 then i could be 50 or 10.but if the second dude finds out his number is 50,then you will find out your 10 and the first dude will figure out he is 20(3 votes)
- In2:42, why does he use ABC, not XYM(3 votes)

## Video transcript

This is another brain teaser
that deals with people trying to logically deduce what's
on their forehead. And it's called the Forehead
Number Game. So what's going to happen is I
guess you could say the god of brain teasers is going
to get three people in a room together. Once again, these three people
are perfect logicians. Which means they have infallible
powers of logic. And they know that the other
two people-- they're three people in total --are also all
perfect logicians and have infallible powers of logic. We get them into the circle,
and we'll say this is the top view. So this is the top of
one guy's forehead. They're all bald for
convenience. To make it easy to paint
things on their head. This time we're not going
to paint blue paint on their forehead. We're actually going
to paint numbers. So you could say the dungeon
master of this game, or the the god of this game says I'm
going to pick three numbers. So three numbers
will be picked. They will be unique. Which means that no two of these
numbers are going to be the same number. There are going to be three
different numbers. They're all going to be
greater than zero. And this is key: One of the
numbers is going to be the sum of the other two. And they tell this to all
of the logicians. So this is the method that they
use to pick the number. And they tell this to the three
parties right there. And what they do is they take
these three numbers. And they write them on
people's foreheads. So say the numbers
are a, b and c. And we know that a plus-- let me
scroll down a little bit b is equal to c. We know that a, b and
c are different. And we know that a is
greater than zero. b is greater than zero. And c is also greater
than zero. What they do is they paint
them on their heads. So this guy will have
a on his forehead. This guy will have b
on his forehead. And let's say this guy has
c on his forehead. And so when this guy looks out--
and you know, the reason we're even putting things on
people's foreheads is because you can't see what's on
your own forehead. No tricks involved here. All you can see is what's on the
forehead of the other two people in the game. So this guy is going
to see a and b. This guy right here is going to
see the foreheads c and a. And then this guy's going
to see these two foreheads, c and b. So that's the rules
of the game. And of course, you don't know;
this guy, the number on his forehead is c. And it happens to be
the sum of a and b. But he doesn't know that. He doesn't know whether
it's the sum or not. He doesn't know he has
the sum number. This is just the way I
happened to draw it. So given those rules
of the game. And let's say you're
one of the players. And this is the view you see. You look at the two other
people in the game. Let me see if I can draw them
a little bit better. So you look at one person. This is your view. So this is one person
in the game. On his forehead-- which
is fairly large-- you see the number 20. And on the next guy, who
I'll draw in purple. The next guy, you see
the number 30. And what happens is-- so this
is literally your view. So they're seeing you with some
number on your forehead. And this guy can
see this 20 and whatever's on your forehead. This guy can see this 30 and
whatever's on your forehead. And the game show host, or the
dungeon master or soon. whoever, says, OK what's
on your forehead? And you look at these two
people, and you say, gee, I don't know what's
on my forehead. I'm a perfect logician, but I
still can't figure out what's on my forehead. And they say, OK fair enough. Then they go to this guy. We call him whatever we
want to call this guy. And they say, hey you,
do know what number's on your forehead? And he looks at whatever
number's on my forehead over here. Let's say this is my forehead. I don't know what number's
written on it. Some number's written up here. He looks at this number and then
at that guy, and says, you know what, I really don't
know what my number is. They say, OK fair enough. Then you turn to this guy. And you say, hey you, look
at this guy and that guy. Do you know what
your number is? And he looks at that number, and
he looks at the number on your forehead. And he says, you know what? I really don't know
what my number is. And then they come
back to you. And then you, as a perfect
logician says, oh, now I know what my number is. So my question to you-- and this
is the question of the brain teaser-- is, what
is your number? And why? So that's the question. And everything I'm going to do
from this point on in the video is essentially
going to be a hint. And then I'll actually
give the solution. So let's think about
it a little bit. And pause it, stop
it, whatever. Think about it for a while. And I would say this
is a fairly advanced level brain teaser. And it takes a lot of
trial and error. But the solution-- just as a
little bit of a hint-- isn't an ultre-complicated one, where
you have to know some higher level math. It's, once again,
very logical. You can make very logical
deductions here. You don't need any higher
math to really come to the correct answer. So let's lay it all out. So if I were to draw
the top view. We'll do it in yellow. This is you. This is the guy with
20 on his forehead. And this is the guy with
30 on his forehead. And what do we know? Let's call whatever's
on your forehead a. That's what you want
to figure out. So we know that a is
greater than zero. These are obviously
greater than zero. You see those already. We know that a can't
be 20 or 30. Now this is the interesting
thing. They told us that the numbers on
one of our foreheads is the sum of the numbers on the
other two foreheads. So either-- this
is a big hint. I think you might be able
to run with this after you get this hint. You say either a is a sum
of these two numbers. So a is equal to 20 plus
30, which would mean a is equal to 50. What's the other option? Either these two numbers, when
you sum them, is equal to a. That's one possibility. The other possibility is 20
plus a is equal to 30. Or, a is equal to 10. And of course I can't say that
30 plus a is equal to 20, because a can't be minus 10. That's the only other
possibility. And we already said that
all the numbers are greater than zero. So every one of these players
knows that their number on their forehead is either the sum
of the other two numbers, or it's the difference of
the other two numbers. This guy says, you know what? My number is either 30 plus a,
or it's the difference between a and 30, depending on
whichever is bigger. And he can see which
one's bigger. I can't see. So you can say the absolute
value of 30 minus a. He knows that about
his number. This guy, he knows that his
number is either 20 plus a, or the absolute value
of 20 minus a. And this all comes from the clue
or the rule that one of these numbers you is the sum
of the other two numbers. Although, you don't know if
you have the sum number on your forehead, or you don't know
if you have one of the numbers that make up the
sum on your forehead. But let's go back to our-- So
right from the beginning of the game, before anyone asks
us a question, we know that the number on our forehead
is either 50 or it is 10. And that's a pretty good hint,
and you might be able to take it from there. I mean, we've narrowed down the
universe of numbers from an infinite number of numbers
to two numbers. And this guy can just deduce
that automatically. So what other information
was given in the game? Well, we actually went
around the circle. So the first time you asked this
guy what number he had, he didn't know. He would be able to say, well
I either have 50 or 10, but that's not enough for me to make
a bold statement that my number is definitely 50,
or that my number is definitely 10. And so what other information
was there? We went to this guy, and
this guy wasn't able to guess his number. Then this guy wasn't able
to guess his number. And now the argument is that
now you should be able to guess your number. So let's think about
what happens. In the situation where this
guy is 50-- Well actually, let's take the other
situation. Let's take the situation
where this guy is 10. If this guy is 10, what
will this person see? And I'll do this situation
in green. This guy will say, I'm
either 30 plus 10. So he'll say, I'm either 40. Or I'm 30 minus 10. Or I'm 20, right? And he says, well, you know, no
real good conclusion I can make there. I could either be 40 or 20,
and I can't figure it out. Fair enough. Now you turn to this guy. In the situation where this
guy is 10, what's this guy going to say? He's going to say I'm
either 20 plus 10. I'm either 30, or I
am 20 minus 10. So 20 minus 10 would be 10. And you say, wait, wait, hold
on a second though. I can't be 10. Because if I were 10 and this
guy were 10, that would violate the rules of the game. One of the rules of the
game said that every number had to be unique. Every number had to
be different. So if you were 10-- you know
from the beginning to you're either 50 or 10-- if you are 10,
then when you go to this guy, he would be able to
deduce that he is 30. And how would he be able
to deduce that he's 30? Because he sees this
20 right here. If you were a 10, he'd
see a 20 and a 10. And he would say, I'm either
the sum of those numbers or the difference. If I'm the sum, I'm 30. If I'm the difference, I'm 10. But then he would say, I can't
be 10, because you're 10. And all the numbers
are different. So I have to be 30. Right? So in the reality where you are
10, when you go around the circle, this guy would
be able to figure out what number he is. Right? Because he's a perfect
logician. But in the problem statement I
gave you, we went around the circle and this guy couldn't
figure out what number he is. He couldn't establish the fact
that he is definitely 30. So given the fact that he
couldn't establish that he is 30, it means that he did
not see a 10 here. He definitely did
not see a 10. If he saw a 10 there, he
would say, I am 30. Because I know I cannot be 10. So we know that you don't have
a 10 on your forehead, and so that's why when you come
back to you, you say, I know I must be 50. 50 is the number on my forehead,
because if I were 10, this guy would have been
able to get it before me. He would have been able to get
the number on his forehead before coming back to me. Anyway, I thought you would
enjoy that brain teaser. I didn't think of this one. I actually didn't think of any
of these, although they have a little bit of my twist
on some of them. But I'll be sure to add more. See