I am a mathematician and I like action movies Particularly Die Hard is one of my favourites You know action movies and there is one which has a couple of maths problems and one very famous one is In this clip, so just play the clip and then we talk about a little bit Careful don’t open it.
What I gotta open and it’s gonna be alright I told you not to open it! I trust you see the message It has a proximity circuit. So please don’t try Yeah, I got it. We’re not gonna run. How we turn this thing off? On the fountain there should be two jugs, do you see them, a 5 gallon and a 3 gallon? Fill one of the jugs with exactly four gallons of water and place it onto the scale and the timer will stop Must be precise one answer more or less will result in detonation if you’re still alive wait, wait a sec I don’t get it you get it
No It’s a problem 30 seconds Mean I’m pretty good at these things But I don’t know if this was the first time that somebody posed this problem. 30 seconds! I don’t know anyway, so let’s just have it really really close look what’s going on here We’ve got two containers – a five gallon container and a three gallon container and now Simon the villain tells us to make exactly four gallons of water just by pouring back and forth between those two containers. No 30 seconds. Well, we probably dead, but anyway Let’s just think about it and see whether we can figure out what’s going on there Alright So here’s a solution, okay? Here’s a solution, and I think anybody who actually thinks about this a little bit will come up with the solution Okay, so what we do is we fill up the five gallon container Then we pour as much as possible from the five gallon into the three gallon container so it leaves us with two gallons in the five and Fills up the other one completely okay? We’ve got two and three there Okay, what comes next well the next thing. We do is. We just get rid of those three? Then we fill the two into the empty one like that, then we fill up the five gallon container Like this, and then we pour as much as possible from the five into this one here well, there’s only one one gallon that fits in here so that gets us to four on that side. Finished, put it on and Well probably ten minutes by now, but you know we’re in heaven. We’re still putting it on Alright now. I’ve got a really really cute Mathematical way of solving this problem and lots of related problems, and I just want to tell you about that one Okay, it goes and actually goes with billiards mathematical billiards. It’s a special sort of billiards table Well if you have a look at it. It’s not rectangular like the billiards table. It’s kind of skewed right so there’s a 60 degree angle here and Well, what are we going to do? Well, first of all have a look at the dimension of this table dimensions kind of a giveaway? You know there’s three units kind of this way, and there’s five units going this way So it must have something to do with the volumes of the containers all right then what we’re going to do is we’re going to put billets ball in one of the corners and then we shoot it at 60 degrees and just see what it does okay, and Well, let’s see what it does right, so I’ll shoot it It bounces here, and it’s reflected off at 60 degrees And it bounces there It goes like that and over here and there and that’s actually our solution How is it our solution? Well? It’s maybe not apparent yet So let’s put in something else so what we think of here is if you think of this and that one here as like axes of a coordinate system like xy Okay, so if this is x and this is y then there’s the origin you know (0, 0) then we go over one This is gets us to (1, 0) then over to point (2, 0) and so on and up here is (0, 3) and they’re up in the Corners well 5 over and 3 up which is (5, 3) ok Okay, so let’s shoot our billiards ball again okay, so we put it there and Right at the [moment]. It says 5 0 and basically what it tells us is 2 fill this container all the way up So we do that Then we [should] Fill its ball like that and Then we’ll just read off the coordinates here two three which tells us that we should Pour water from here to there as much as possible Because there we go Alright now it gets reflected off down to there. So we’re now a (2, 0) which tells us We should get rid of the water in the three gallon containers. We get rid of that then follow it further, (0, 2) that means we’re pouring the water from left to right and Then we go over here. Which just means we’re filling the 5-gallon up all the way up and Then one more step and that tells us well Pour some water from the five-gallon into the other one and that leaves us with (4, 3) It’s exactly the solution that we came up with at the beginning so pretty neat Okay, well What if Simon had asked us to put one gallon of water on the scales? well then that Method he actually tells us what to do because what we do is. We just kind of Keep on going in keep on going so it’d be the ball kind of keeps on bouncing here And let’s just go all the way and you can actually see it goes across all the lines that I had here originally okay, and Well now if you have a really close look you can actually see that in this position here that Corresponds to one gallon being in here and nothing [being] in there, so you just take this this container for little scales And you’re okay now This one here corresponds to two gallons. So it could also have done two gallons if he had said two gallons No problem this one here says three gallons pretty obvious you can just fill up that one and put it on so it Would be a really easy puzzle so I mean Simon’s meaner than that Four gallons we can also do we’ve just done that five gallons easy But there’s more you can also see at this point in time There was five gallons in here and one in there So we could actually take both containers put them on a scale And that would be six gallons of water right and in this position here We have five gallons in here and two gallons in here. So that would give us seven gallons We could put that one on solve another puzzle And we could also do five plus three of course, eight. So all the numbers from one to eight We can actually do as this diagram shows us Okay, there’s more in here. I Was wondering actually? then isn’t there a Faster way to get to one as soon as we touch by one or one three without having to complete the graph Yeah, like so as soon as as soon as you touch as soon as you touch anyone you’re done. Okay? Touch anyone and you’re done. That’s fine. Yeah so there was my My friend colleague and cameraman giuseppe who just had the right thing to say All right now next thing is There’s lots more hiding in this diagram, and then you would imagine okay, so there’s a second solution here and the second solution Well, how does this come about well? You could actually start from here and start from there? Why don’t you just shoot the ball from there, and then what happens well? Let’s see (0, 3) Corresponds to filling that one up first remember before we filled that one up And then what kind of the water was kind of flowing in this direction You kind of always fill this one up and pour over there until it’s full and then you kind of get rid of stuff Right and now kind of the waters flowing in the opposite direction okay, okay? Now. We shoot the billiards ball down there Tells us transfer the water alright now. We shoot it up. There tells us fill that one up water is flowing from here, huh? Okay, next one five one so we pour as much as possible over here Gets us to five one okay, then What are we supposed to do now? Just get rid of that one no then? Transfer the one then fill up the three then You know just pour it in and you’ve got the other four. That’s the second solution It’s a bit longer than the first one, but also works pretty neat right Okay, what else is there? Well? I should really tell you why this works all right? Yeah, I’m the mathologer I’m not really happy about all this stuff Until I’ve got a really really good explanation for these things Okay, so how do we go explain this? Basically the method works because the individual bits That the path made of work. What I mean by this – let’s have a look at one of those connections here What does that connection actually stand for well it stands for? one of the jugs being completely empty the (5, 1) and the other one has something in it and then what this Connection stands for is Just filling that guy Which is empty all the way up and if you go the other way which might also happen it Just means you just emptied a completely full 5 Gallon Jug you know that’s it right and obviously if you’re in a in a state like this, if you’re in state (0, 1) you can go this way and if you’re in state
(5, 1) you can go there So this connection here stands for something that really works in reality, okay? That is the second kind of connection here, and that’s this one here It’s exactly the same sort of thing instead of filling an empty five Or empty a full five you’re now filling an empty three or emptying a full three okay? So exactly the same thing this sort of connection here corresponds to something that really works in reality, okay? I’ve got a third kind of connection. It goes this way Well as if this actually will become more complicated here the situation because well, you can have a connection like that You can also have something like this And it can be done there and there. So you know, what what’s going on here? Well for this one here to really see what’s what’s going on. It’s actually helpful to kind of go in little segments here so let’s start from here and just Go to this point here Okay, that’s actually three one, okay So it’s three over one over. So what what are we actually doing when we’re kind of moving along this this segment? well, we are kind of Pouring water out from the from the big one from the five gallon one and we’re filling it into the other one, right? So here we start with (4, 0). We’re going one down and one up total amount of water stays the same right and So we’re moving in this direction. We’re basically pouring water from here to there One segment here corresponds to one gallon that is busy making a transfer, and you hit the other side Well, you got exactly the right amount you can basically hitting this boundary so it’s going to work all right so it’s going to work This sort of transfer again Corresponds to something that you know happens in reality So these these two points when you actually connect them You know that that really works. Now the path of the billiards ball it’s just made up from these individual segments that work, right? so if we start out with something that we can actually achieve like a (5, 0) Then all the other bits are connected by a things that works or the whole thing has to work So we automatically do the right thing nothing can go wrong okay, what else? Well, at some point in time there’s going to be Die Hard 25 and Actually actually pride myself that I’m a little bit I looked quite similar to Bruce Willis So I’m going to see whether I can be Bruce Willis in Die hard 25, and then solve the problem with my method here so, but in Die Hard 25, there’s also going to be a Simon’s going to be back as out of prison by now. I don’t know actually if maybe he died, he probably died… You can check it? Okay Giuseppe is going to check it. He’s gonna check it on his iphone while I keep talking Anyway, so let’s say Simon’s back either from prison or from the dead, and he’s going to tell me well this time I’m going to make it harder for you I’m going to have a 6 gallon jug and a 15 gallon jug and what I want you to make is a 5 gallons of water Okay and I’m going to you know see whether my method actually works, right? So what I’m going to do is I’m going to make Up a billiards table that has the dimensions 15 and 6 and then I’m going to just run my ball again And I know well Fingers crossed and see what happens when I run my ball and actually I mean there’s a bit of a problem here because the Biltz ball actually doesn’t hit every single one of the points down here, so it doesn’t hit any one of the Doesn’t hit the one, doesn’t hit the two, it hits the three, not the four and not the five, hits the six and in fact All the coordinates that you come across here are multiples of three and That’s sort of true in General, so if you’ve got you know volumes here and there then What you can do is… what you have to do – is you basically take the greatest common divisor of the two numbers which? In this case is three and then all the volumes you can make up are just multiples of of the greatest common divisor Well, in this case we’re actually out of lucky. He said I was supposed to do a five account to a five There’s no, you know There’s no five anywhere in here that that’s going to work, so I actually have to think of something else I’d probably be dead 30 seconds. I won’t be able to think of anything else