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Current time:0:00Total duration:7:25

Blue forehead room brain teaser

Video transcript

there's a new reality television program and it's called the blue I should probably write it in blue but it's called the blue for head room blue for head head room and what they do in this reality television program and and you'll have to bear with me because this show probably wouldn't be that interesting to watch but it's interesting to predict what happens what they do is they take a room we'll call it the blue for head room and they let's see that's that's kind of a top view of the room and let's say there's a door here none of this is relevant to the actual problem this is the door right oh right there and what they do is they get 100 perfect logicians to sit in this room in a circle so they're all you know sitting in a circle in this room now before the game even starts before they even enter the room the first time the logicians are told two things they're told one that at least one of you at least one of you has your forehead painted blue has your forehead painted blue at least one of you has referred and maybe you know that they all get their foreheads painted so that you know obviously if you're the only guy who has your forehead painted and you know but you don't know what color it is so all of them have different color foreheads right or we don't know but all they're told is look you know obviously I've painted your forehead and at least one of the people in the room that you will enter will have their forehead painted blue and then they're also told that as soon as soon as you deduce that your forehead blue that your forehead is blue that you're blue you need to leave the room you need to leave the room and what's going to happen is and it's it's very important that I set this up properly they're all outside of the room no one's inside the room and let's say they're blindfolded and while they're blindfolded they essentially you know have the thing you know painted onto their forehead so they can't see the paintbrush or anything so really really don't know what's on their forehead and they just have to use and and then after that they're essentially allowed they all enter the room right and they all sit in a circle like this so that they can all look each other then the foreheads and then the and then the and let's say when they enter the lights are off right so the lights are off and then what the the the protocol is is that the lights will be turned on and then they can all look around at each other there's no reflective surfaces they can't look at it you know look into each other's eyes and try to see the reflection no tricks like that there are no mirrors in this room nothing like that all they can do is look at each other so let's you know just as an example let's say that this is me right here as soon as the lights get turned on the first time I'll be able to look at all the other people in the room and I could see really it'll be pretty obvious to me if anyone has a blue forehead right eye maybe that guy has one that guy I don't know right and I could see them I can't see my own forehead and what happens is it then they will turn off the lights and the way they're going to do it is you have to leave the room after you have realized that you have a blue forehead so for example let's say I enter into the room and you know because I'm a perfect logician I see things that make allow me to perfectly to do deduce that I have a blue forehead then what they're going to do is they're going to turn off the lights again and then if I know that I have a blue forehead while the lights are turned off I would leave the room and then when they turn the lights back on I'd be gone so there would be no Sal here so you know let's say they're 100 before then there would be 99 guys sitting in the room right as soon as I I as soon as I realize I have a blue forehead when the lights get turned off I leave right and just remember these are perfect logicians so everyone in the minute and and and not only are they all perfect logicians but they all know that everyone else is a perfect logician so everyone is also told and this is true everyone everyone is perfect logician which means they have infallible powers of logic perfect logician logician so my question to you so you know just remember I have you know each of these perfect logicians we set them up outside of the room paint their foreheads they're blindfolded they have no clue then we have all of them walk into a dark room sit in a circle like this and then what we tell them is as soon as you realize that you have a blue forehead as soon as you have a blue forehead you have to leave the room now my question to you is let's say that we've actually painted everybody's forehead blue what happens so remember we've this is what we've told each of the people right as soon as you realize your forehead is blue you leave the room and now I've just asked you what happens and you know the producers like to really play with these logicians they've actually painted everybody's forehead blue so you know you know when when everyone goes to the room the first time what's going to happen you know let's say this is let's say I'm one of the logicians this is me right right here as soon as I open my eyes I'm going to see 99 other fellow logicians with blue foreheads so that's that's just kind of and then you know maybe I can somehow to do something about my own not you know I don't think you can and then the lights will go off and then you know if I haven't deduced anything about my own blueness of my forehead then they'll turn the lights back on and then maybe some other guy will have left I don't know or or maybe not and then I'll see the same 99 guys again and that'll just keep occurring until something happens and my question to you is what happens when and why and that's the well I'm thinking whether I should give you a hint right now well let me tell you this way that's the problem you should be able to solve it and just you know where this came from this side if I remember correctly I think this Kate was on a computer science exam I had at MIT so it's a you know just so you know this isn't just for fun this is this could actually you know this well I I don't want to I don't want to go into the into the into all the applications that this type of problem can apply to because that by itself would be a bit of a hint but let me so if you don't want a hint turn off the video now or pause it if you do want a hint well I'm about to give it to you so my hint is my hint and remember turn this off if you don't want a hint but the hint is what happens what happens when there are less than 100 people so you know I gave the the situation where we have a hundred logicians but this problem is actually a lot easier if you try it with a smaller number anyway I'll see you in the solution video