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:11:00

Blue forehead room solution

Video transcript

so we had the hundred logicians all of their foreheads were painted blue and before they entered the room they were told that at least one of you at least one of you hundred logicians has your forehead painted blue and then every time that they turned on the light so that they could see each other they said okay once you determine that you have a blue forehead when the lights get turned off again we want you to delete leave the room and then once you know that's kind of settle down they'll turn the lights on again and people look at each other again and then you know they'll turn them off again and maybe people will leave the room so far or it's so forth and so on and they're also all told that everyone in the room is a perfect logician they have infallible logic so the question was what happens and actually maybe a even interet more interesting questions why does it happen so I'll answer the first what happens and this if you just take the answer you don't know why it almost seems mystical is that essentially the light gets turned on and off 100 times and then after the hundredth time that the light gets turned on so 100 times after the hundredth time that the light gets turned on and the lights get turned off again all of them leave they all lead they have after F so I mean it's kind of weird right we let's say I'm one of them or you're one of them right I go into this room the lights get turned on and I see 99 people with blue foreheads and you know there's I there's i don't can't see my own for it they see my forehead of course but i'm one of you know to each to any other person I'm one of the 99 right but I see 99 blue foreheads so essentially what happens if we were to watch this show is the lights get turned on you see 99 blue foreheads the lights get turned off again then light and then lights get turned on again and then all over one's still sitting there and I still see 99 blue foreheads and that happens that happens a hundred times and after the hundredth time the lights gets turned on everyone leaves the room and you know at first at a kind of first glance that seems crazy because nothing changes right nothing changes between every showing of the I guess every time we turn on the light but the weighty need to think about this and this is what makes it interesting is what happens instead of 100 let's say there was one person in the room so you know before the show starts but sat you know they never told me that they're going to be a hundred people in the room they just said at least one of you at least one of the people in the room has your forehead painted blue and then as soon as you know that your forehead is painted blue you leave the room and that everyone's a perfect logician so imagine the situation where said of a hundred there's only one perfect logician let's say it's me so that's the room I walk in and I sit down and maybe I should do it with the blue and then they turn the lights on and say look at look around the room and I look around the room and I see nobody else right and remember even in the case of one we've painted everyone's forehead blue so in this case this one dude or me or whoever you want to call him his forehead is painted blue so he looks around and and he sees no one in the room but he remembers the statements and maybe it's even written down on a card form and in case he forgets that at least one of you has your forehead painted blue so if he looks around the room he says well I'm the only dude in the room and they told me that at least one of the dudes in the room it's going to have the foreheads blue well I'm the only dude in the room so I must have a blue forehead so as soon as they turn the lights off he's going to leave alright fair enough that's almost a trivially simple and and today you might say well so how does this apply to 100 right well what happens when there are two people once again both of them have their foreheads painted blue so let me draw another I don't want to keep drawing the the blue forehead room let's say there's two people now so right when the so let's put our hit our selves in the heads of this guy right right behind the blue forehead that's where we're sitting okay so when he enters the room when you know he says I either have I either have a blue forehead I have an appointment meeting let me cancel that I either have a blue forehead or I don't have a blue forehead no blue right this is this is what this guy is thinking let me draw him and he has a blue forehead but he doesn't know it he can't see it that's the whole point about painting the forehead blue as opposed to another part of the body so he says I either have a blue forehead or I don't have a blue for it he walks in right he walks in let's say this is this guy he walks in the first time the lights get turned on he sees this other dude there who has a blue forehead and he says okay now let me think about it how will this guy respond depending on each of these states so let's say that I don't have a blue forehead let's go into this reality if I don't have a blue forehead what is this guy going to see well when the lights get turned on he's going to see that I don't have a blue forehead and we were both told that at least one of us has a blue forehead so this guy because he's a perfect logician will deduce that he has to have a blue forehead remember this is in a situation if I assume that I don't have a blue forehead we're in this world right I'm a perfect logician so if i can assume if i if I'm kind of a simulating the reality where I don't have a blue forehead then this guy will see I don't have a blue forehead and then he'll say I must have a blue forehead and so when the lights get turned off this guy will leave the legs at the room and you know vice-versa that the other guy will make the same logic but let's but since both of them have blue foreheads right what happens the lights get turned off then the lights get turned back on when the light gets turned back on this guy is still sitting here right and I just determined that if I didn't have a blue forehead and this is me if I didn't have a blue forehead this guy would have left right because he could have he could have said oh I must be the only guy with a blue forehead so he would have left but since he didn't leave I now know that I have a blue for so then the second time that the lights get turned off or the after the second the second time that the lights get turned on I can deduce that I have a blue forehead and then when the lights get turned off again I'll leave and this guy he's a perfect logicians so he makes the exact same conclusion because he also kind of simulated and said okay if I didn't have a blue forehead this guy would say if I don't have a blue forehead then this guy will leave as soon as the lights get turned off the first time right if this guy doesn't have a blue forehead then this guy will say well I'm the only guy in the room with the blue forehead well I see no one else in the room with the blue forehead and since I know that there's at least one with the blue for it I'll have to leave but since he didn't leave this guy will also know that he must have a blue forehead as well so they'll actually leave together maybe they'll bump each other bump into each other on the way out fair enough now what happens if you extend it to three people so we already said if you have one person in the room he'll leave the first time the door he'll come to the conclusion the first time that the lights are turned on and then he'll leave right when there are turned off if you have two people it takes them essentially two times for the light to get turned on to reach that conclusion now if you have three people if you have three people and I think you see where this is going you have three people one two and three and remember no one knows if their foreheads are painted blue but the producers of the show actually did paint everyone's forehead blue so this so once again let's get into this guy's head so this guy says he's either blue or he's not blue not blue so in the reality when he's not blue what what's going to happen well this guy and this gets a little bit confusing but if you think about it from the previous example it makes a lot of sense right a not blue person actually doesn't it a person who has a not blue forehead actually shouldn't affect the outcome of what the blue people do because let's say that this guy says well what's going to happen if I'm not blue well these get this guy is going to look at that guy oh he has a blue forehead he doesn't have a blue forehead so if I don't have a blue forehead this guy is going to see two people without blue foreheads right and he's going to leave the room on day one right but then let's say or not day one but the first time that the lights are turned on he'll come to the conclusion now the second time the lights turned on this guy will say gee this guy didn't leave the room that guy doesn't have a blue forehead and this guy didn't leave the room because he must have seen someone with the blue forehead therefore I must have a blue forehead and so if this guy doesn't have a blue for it both of these guys would leave the room the second time that the light gets turned on now what happens if they don't leave the room the second time that the lights are turned on well if I was not blue they would have left so if they haven't left by the third showing of the light then I know that I am blue so when you have three people they're all perfect logicians they all have their their foreheads painted blue on the the the light will be shown three times or the the light will be turned on three times and then when the light gets turned off they'll all leave together they'll all leave together and so this logic applies for any you could have a thousand people you could keep extending it the fourth person will have the exact same logic will have the same logic if he has if he has if he's not blue if he's not blue then these guys are going to leave after three showings of the light or three turnings on of the light but they don't leave after three turnings on of the light then he must be blue and so they're all going to leave together all four of them on the firt on the fourth showing of everyone's foreheads anyway and so you could keep extending this all the way to 100 and 100's arbiter you could do this a million people and then they would just keep looking at each other a million times and then of a millionth showing they would all reach the conclusion that they all have blue foreheads and they would leave the room and then you know if you think I mean it's it's fairly straightforward logic but it kind of come it leads to kind of a very almost eerie result hopefully that that satisfies you see in the next video