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Forehead numbers brain teaser

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

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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