If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:12:09

this is another brain teaser that deals with people trying to logically deduce what's on their forehead it's called the let me see the forehead or head number game number game so it's going to happen is the I guess we 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 so they're a different color there are perfect logicians which means they have infallible powers of logic and they know that the other people the other two people that are the earth are the three people in total are also all perfect logicians and have infallible powers of logic we get them into circle let's say this is a top view so this is you know the top of one guy's forehead they're all balled for convenience to make things 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 the 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 they're going to be three different numbers they're all going to be greater than zero and then this is key one of the numbers is going to be the sum of the other two so one number sum of other other two and they tell this to all the logicians so this is the this is the method that they used 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 you know let's say the numbers are a B and C and we know that a plus let me scroll down a little bit we know that a plus 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 they paint them on their head so this guy I'll 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 at you and you know the reason why we're even putting things on people's forehead it's 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's going to see a and B this guy right here is going to see the foreheads C and a and this guy's and then this guy's going to see these two foreheads C and B so that's the the rules gave him of course you don't know like you're 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 the sum number right this is just the way I happen to draw it so given this rules game but let's say you're one of the players right you're one of the players and this is the view you see right 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's this is your view so this is one person in the game and hit on his forehead which is fairly large you see you see the number 20 you see the number 20 and on the next guy who I'll draw in let's see maybe the purple is good color the next guy you see the number 30 and what happens is is they tell you so this is this is literally your view so you know they're seeing you with a forehead some number on your forehead this guy sees you with an this guy can see this 20 and whatever is on your forehead this guy can see this 30 and whatever is on your forehead and the I guess the the game show host or the dungeon master whoever says okay you what's on your forehead and you look at these two people and you say gee you know I don't know what's on my forehead I'm a perfect logician but I I still can't figure out what's on my forehead and then they say okay fair enough then they go to this guy right we could call you know whatever we want to call this guy and they say hey you do know what numbers on your forehead and he looks at whatever numbers on I forehead over here let's say this is my forehead I don't I don't know what number is written on it something that some numbers written up here he looks at this number and then that guy says you know what I really don't know what my number is then you say okay fair enough then you turn to this guy they 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 four 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 the question of the brainteaser is what is your number and why what is your number and why so that's the question and everything I'm going to do from now 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 this is a you know I would say this is a fairly advanced level brain teaser and it takes a lot of trial and error but the solution just is a little bit of a hint isn't like some ultra complicated one where you have to know some you know some higher-level math and and and you know you have to it's it's once again kind of a very logical you you could you can make very logical deduction here and you don't need any higher math to really come to the correct answer so let's let's let's lay it all out so if I were to draw the top view we 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 we know whatever 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 twenty or thirty now this is the interesting thing they told us that the numbers on one of our foreheads is the sum of the numbers of the other two foreheads so either so and this is this is an interest this is a big hint I I think you might be able to run with this after you get this hint you say either a is the sum of these two numbers so a is either 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 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 right 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 right this guy says you know what my number is either is either 30 plus a or it's 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 because so you could 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 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 for it but let's go back to our you know 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 now I'm actually 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 so and this guy can just deduce that automatically and 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 I mean he would be able to say well I either have 50 or 10 but that's not enough for me to make a you know a bold statement that my number is definitely 50 or that my number is definitely 10 so what other information was that 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 now you should be able to guess your number so let's think about what happens in the situation where this guy let's say the situation where this guy is 50 well actually let's take the other situation let's take this situation where this guy is 10 right if this guy is 10 what will what will this person see and I'll do this situation in green this say 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 they say wait 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 the one of the rules of the game said that every number had to be unique every number had to be different so if this guy were essentially you if you were 10 right you know from the beginning that 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 do since 30s because he say well I'm either the I you know he sees this 20 right here and if you were 10 he would see a 20 in a 10 and he would tell me there's a some of those numbers or the difference if I'm the sum I'm 30 if I'm the difference I'm 10 but then you 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 this guy couldn't figure out what number he is he couldn't establish the fact that he is definitely ten right so given 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 right 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 be able to get the number on his forehead before coming back to me anyway I thought you would enjoy that brain teaser it's a it well anyway I'll try to think of well I didn't think of this one this was actually given to a 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 you soon