 All right, so recently I watched an episode of Brooklyn Nine-Nine and it had Have this puzzle in it and immediately I decided to stop watching and just try to solve the puzzle it took me a few days it took me two days and You got to think a little differently After after you hear so The puzzle is That There are 12 men on an island 11 way the exact same amount But one of them is slightly lighter or heavier You must figure out who it is and whether they are heavier or lighter by using the seesaw the catch is you can only use it three times all right, so first we got to look at the possible the possibilities and So there are 24 possibilities right now, so Out of the 12 men they could be lighter or heavier and so they're and so 12 times 2 is 24 All right, so the first thing you got to realize is that when you do a scale when you When you compare two groups you have to The most amount of information you get is reducing the possibilities by division by three so So that's the most amount of information you can get and If you realize this you got a You know that the optimal that the optimal first move is Going to be All right is going to be comparing four and four and so I'm just going to compare these two Right here and if it is left if it leans towards the left Then we know that if the guy is within these four then he has to be heavier And if the guy is within these four it has to be he has to be lighter So we've reduced the possibilities to eight If it leans rightward then it's flipped all right But if it is if if it goes to the center These could either be positive or negative so there are still only eight Possibilities all right. Oh, that's by the way and by maximum amount of information You can get is I'm talking about worst-case scenario here. So if you were to come to Compare one and two you could get You could reduce it to one or two but that But the amount of in but in the worst-case scenario You still don't have you still don't have a lot of information. Okay so So now we realize that if we compare for four and four we get eight possibilities, right and We realize that if you think about it, we realize that by the second Turn by the second comparison you got to reduce the possibilities to three because you can only get left That's not an L. You can only get left center and right And so we got a so we can only find out if there are only three possibilities all right, so let's start with a scenario where Where these are positive and this is negative and If it's the other way around then it's just you just flip the signs when you do it like this So since we know we got to reduce it to three We know that the amount that goes unmeasured has to be three So we have to have five that we compare all right and I'm gonna be honest I tried randomly Combinations of five. Oh, yeah, we also know that it has to be asymmetric. I don't know That's just what I intuitively think. So here's the comparison you go one two Five six And then I'm gonna use a triangle for as like a comparing symbol five six seven nine ten Eleven all right if it leans rightward Then that must mean that either one or two are Greater than or that seven is lighter I don't know why I said greater than either one or two are heavier or seven is lighter if it leans towards the center then what are the ones we didn't compare we know four has to be pop has to be heavier and we didn't oh, yeah, we know sorry We know three and four have to be heavier or eight Or eight has to be lighter. All right, so that's the rightward case Center case Well, no, this is the leftward case. Sorry. This is the leftward case. All right and In the case that it leans rightward then we know that either Either five or six are lighter or Let me think about this Wait, sorry, there is no or it's either five or six are lighter So let me get rid of that. All right, and from here. It's sort of trivially easy to find out What's what so I'm just gonna do the quick Do this one real quick. So in the case that it's leftward the next the next comparison you want to do is two or two and seven and compare it to nine and ten because we know that these are the These are the same If it leans rightward, we know that it's two because that's the heaviest heaviest one If it goes to the center, then we know it's one because that's the We know that one is heavier since that's the only one that we didn't measure and if it Leans up. Sorry if it leans rightward Then it means that seven is lighter if it leans leftward and that means that two is two is heavier And if it's in the center, we know that one is heavier. All right We do the same thing for this one except, you know, we change the numbers and for five and six we Let's we measured we'd compare five to one to like nine and If if it leans if it leans rightward, then we know Then we know it's five and if it's center, then we know it's six All right, and in the case that these signs are flipped. So all right in the case that This the seesaw goes in favor of five through eight Then these signs are just flipped. So this would be positive. This is positive This is negative. I mean lighter This is seven is heavier. This is lighter and we do the same thing All right So now in the case that they are Equal then our possibilities go from nine ten 11 and 12 These are either heavier or lighter and We would need to compare nine ten. Sorry about the noise. We need to compare nine and ten 11 and Let's say one all right in the case that nine and ten In the case that it leans leftward, then we know either nine or ten have to be positive or I mean heavier or 11 has to be lighter Okay, if it leans towards the center if it doesn't lean anywhere, then we know that 12 There are two possibilities now that 12 is either Heavier or lighter If it leans towards the right Then it's the same thing up here except the signs are flipped nine and ten would either be lighter or 11 would be heavier and Basically, we do the same comparisons as I showed before But this time with 12. It's simple compare 12 to one that we already know Let's do one again. If it's heavier if 12 is heavier, then it's 12 heavy if it is If it leans the other way, then it's 12 lighter Yeah, that's it. I just wanted to show my solution to that