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# Heavier ball

## Video transcript

so we have these nine balls right over here we're going to assume that they are completely identical at least they're identical and appearance but one of the nine balls is heavier just a little bit heavier is heavier than the other eight balls and my question to you is what's the minimum number of times that we can use this scale in order to know definitively which is the heavier ball so there's some number of weighings using the scale that after that number of wings I know for a fact that I found the heavier ball we're not going to do something based on luck that you just happen to pick the right ball when you weigh it it has to be a hundred percent chance after this number of wings that you have the ball and so what is the minimum number of those what is the minimum number of wings of using this scale and I encourage you to pause the video and think about it as long as necessary to come come up with your own conclusions so I'm assuming you've given a go at it so I'll give you a couple of hints now so my first hint is that you can do it in exactly two wings of the scale if I do two wings of the scale I know for a fact that I can find that not just through luck I can find definitively the heavier ball so that's my first intent if that helps you out pause the video I'm about to give you another hint so my second hint is that with each weighing of the scale you should be able to rule out two-thirds of the balls that are essentially still candidates for the heavier ball so if that helps you once again pause the video so now I'm assuming you had a go at it and maybe you were able to figure it out maybe you weren't so now let's work through it together so I mentioned that in each weighing you can rule out two-thirds of the ball so how do we do that so in the first wing what we're going to what we essentially do is take our nine balls and and put it into into three groups of three and we take two of those groups of three so we take this group let me actually do that in a different color so we can take this group of three right over here put those three balls on that side of the scale and then we could take these three balls and put it on that side of the scale and so you're essentially weighing three versus three be balls now there's a couple of outcomes here you're either going to have a balance you're going to have the left is heavier so let me write that say it's going to tip down so the left the left is heavier or the right is heavier or the right is heavier now what does each of these tell you well if this if they balance that tells you that this third group has the heavy ball so actually let me write it this way if this is Group 1 group 2 group 3 then this tells you that group 3 group 3 has heavy ball has heavy ball if the left is heavier then we know Group 1 has heavy ball has a heavy ball Group 1 has the heavy ball and then finally of course if the right is heavier we know that group 2 group 2 has the heavy ball now just like that with one way we have narrowed it down to one of the three groups we've essentially narrowed it down we not now know that our heavy ball is one of three balls it's either one of these three one of these three or one of these three and so I just repeat the process but instead of doing it with with with three balls at a time I not do it with one ball at a time so if I'm taking three balls if I have three balls what I could do is I will now so my step two I guess I could say my step two I now weigh one versus one and once again I have the outcome so if they are balanced if they are balanced then that means the so once again if we're taking say this group of three we're in the balanced situation from the first way and so if we put if we put this ball here and this ball here if they are balanced then we know that this must be the heavy ball because these two are the same if if the Left goes down if the Left goes down then we know this is the heavy ball and likewise if the right goes down if the right goes down we know that this is going to be the heavy ball so this is actually a little bit of a brain teaser that you see it's a pretty common one it's actually even sometimes you'll hear it in some job interviews but you could see it comes drought of the idea that through each weighing you can rule out two-thirds of the balls and so you could use this principle if you want to derive other brain teasers what if you had 27 balls how many wings would you need what if you had 81 and sometimes when you see this brain teaser instead of giving you a nice clean I guess you could say power three right over here they might give something off so they might give you eight balls but the exact same principle holds if you had eight balls you could you could split it up into two groups of three and then two more and then do the same and then do the exact same process but I think people like to do the eight balls because it takes you a little way from the idea maybe you have to divide it into groups of three or something like that anyway hopefully you have enjoyed this