The Return of the Legend of Question Six - Numberphile

this is going to be awesome I'm glad you're here all right so would you like to know how to solve question six yeah well I'm going to show you how to attempt a problem like this but then I'm actually gonna give you a sketch of how the most awesome way to prove it is say it was just me doing this right doing a problem like this and if you look at a problem like this and it's really hard we know the setup if you don't have any idea you've just got to plug in numbers all right you just got to put in the grunt work you've got 90 minutes as we use it we've got a squared plus B squared on a b plus 1 the problem with an equation like this is that somehow there has to be a proof okay and it is totally mysterious at the moment what it could be so what you want is to actually look at what answers the suitable answers what they actually look like you just want to see the Machine working before you think about taking apart know how it works yes but I want to see the machine producing this thing because if it produces a fraction it's really nothing to me okay but if it produces a whole number I want to see if that's going to give me some insight so I'm going to try putting in a equals 1 B equals 2 it's safe for small numbers so if we plug them in we'll get we'll get 1 times 2 plus 1 so that means we're going to get 5 on 2 sorry no no it's right you're 3 yep went too fast there so basically it was 1 times 2 2 plus 1 yeah yeah that's right ok thanks for that help it's a point already yeah it is but also it's it's who cares it didn't really tell me anything so I'm going to try something else maybe let's try something else ok maybe I'll go eyyy equals 0 B equals 2 all right remember we can choose 0 so now let's throw them in so we get 0 squared on 2 squared 0 times 2 plus 1 this is looking promising so that means it's now 2 squared over 1/4 square and I'm not even going to work out what 2 squared is because it's actually in the form I want it now 0 and to actually produce a whole number which is equal to 4 but more importantly it's a square number what's great about this is if I have a look at the way this is this is actually happen if you have a look at the bottom line there it's 0 times 2 0 squared plus 2 squared it didn't have to be 2 I mean that would work for 3 you have 3 squared + 0 times 3 0 down there is going to wipe out whatever is there it just leaves the one on the bottom so 0 times 3 0 0 4 0 5 0 so natural fact I found an infinite number of solutions I've got 0 1 0 2 0 3 0 4 ball and I already know that they're going to produce 1 squared and that one there is going to be 2 squared this one's going to produce 3 squared is going to be 4 square so there you go pretty good huh so we know every square number will fall hours question yep not only that I found infinite number infinite infinitely many solutions I mean that's pretty cool is it the only problem is that's not enough even haven't having infinitely many solutions isn't enough if a pair of integers are thrown in there and somehow it was like a number like 30 that's not a square number and then you just you haven't proved what you need to prove the other thing is they're telling you this is the case so this is when it gets really hard when you have this realization and the smarter you are the quicker you actually realize this it's actually at this point you hear you hear something laughing in the distance and it's the problem it's laughing at you because it knows something and it's not telling you anyway so now we've now we've got to try and do something else and now the problem is is that if you don't have any ideas you are forced to just do grunt work I'm going to quickly do that here actually I'm just going to put in like a matrix I'm going to go 0 1 2 3 4 5 6 7 8 so these are my eighth and I'm going to go down is I'm going to go 0 1 2 3 4 5 6 7 8 now I'm going to start plugging in these numbers and here will be the result of this can I just emphasize I have no reason to expect this will work but if you don't have any ideas and if you can't do the algebra playing around with algebra sometimes it's what you're resorting to do so first of all these are all good so this one here produces 0 1 squared 2 squared 3 squared 4 squared 5 squared 6 squared 7 squared 8 squared so blue means a winner we've already found them 2 0 1 0 2 yeah okay and we also have 1 1 as a winner as well all right so I only have to populate a diagonal because like we're repeating here so I'm going to quickly work out these fractions and so just imagine under exam conditions what you need to do here so you need to kind of look at these things and you go yet ok yeah ok that's no good there yeah mmm-hmm geez Louise this is this is pretty hard so you know your emotions might be going up and down at this point you're sort of sweating you know you're playing for your country here what's happening huh whoa hang on is 8 - is that right did I get 4 I think it is I think its port hang on I have to write it again 4 hang on yep that is yep we found another one so after all that after doing all these calculations and getting nothing we've got an answer we've got an answer with 2 & 8 so now the question is why so let's actually throw it in there so if we actually put in 2 squared well the thing to realize here is that 8 is 2 cubed yeah so now if we put in 2 cubed there and then we have 2 times 2 cubed plus 1 so if we do a little bit of algebra we end up finding that this gets to be 1 plus 2 to the 4 and the bottom is 2 to the 4 plus 1 so that actually cancels with that and we get a secret solution so now we've found two and two cubed also generate another whole number and it's a square right but just like before we use twos here but it actually will work the same you can run the same logic with threes or fours fives any numbers whatsoever whole numbers so we've just found another infinite number of solutions we've got two and two cubed will generate two squared and we have three and three cubed will generate 3 squared and 4 and 4 cubed so really this is amazing so in actual fact we found another infinite number of solutions and it took us a lot of effort so now we feel confident we've got all of them except we haven't and this is the genius of this problem is that there is a VIP super solution which you have to be very clever to find out ok do you want to see it yes this is the reason why this problem is so hard because not only you doing grunt work is going to help you get to the secret solution secret the amazing secret solution to solving this problem is actually a beautiful piece of observation it is I'm going to show it to you right now what the people who solved it realized was that out of the solutions that were semi easy to find there were kind of groupings together of answers that were the same so 0 & 2 produced 4 and so did 2 & 2 cubed right & 0 & 3 produced 3 squared etc right so the very smartest of the people that actually were able to solve this problem kind of worked out I wonder what the connection between these pairs are right so what they did was they actually looked at this equation so they looked at the equation which is a squared plus B squared they looked at the equation for a squared plus B squared on a b plus 1 and they looked at it just for one of those solutions so I'm going to choose 4 and so now what they did was is they actually graphed they graphed solution so here we've got a and here we've got thee okay so now they looked at the two solutions they had 0 & 2 if you stuck that in there you get 4 so that's a solution 0 & 2 and also 2 & 8 for another solution that gave you the 4 so that means that we could draw a graph so let's do it like this so if we say a equals what is it equals 0 and B equals 2 okay okay so there's that solution then and then we know another solution is at 2 8 so that's a equals 2 and equals 8 so that's up there and then you know you kind of like draw your graph like this and you kind of wait draw a graph that goes like this and actually cuts through here so now if we did this on like on my grid paper we're only interested where this actually intersects the grid because it's like where it's a whole number and a whole number but what's amazing at this point here this is 0 2 and this is 2 8 there's actually two more solutions that we haven't included and this is the most beautiful thing about this problem because it actually comes down to a simple observation if we look at this equation here if we swapped around a for B what would we get well if we write instead of a we write B and instead of B we write a and instead of a will write B and that that equation is the same it's just rearranged you can't always do that some equations if you if you move around the variables it changes it but in this case you move them around it looks the same so that means we've actually got two more solutions 0 2 there's also 2 0 there's actually a solution there and there's also another solution at 2 8 and really to draw this graph properly it actually splits into something I didn't realize when I first looked at it so now this is nuts because these two lines represent this one equation so what does that mean well it means if we actually look at - I haven't lined this up properly here but this is supposed to be for a equals two there's actually two solutions for a equals 2 B equals 0 and B equals 8 so we've actually found a link between both solutions so if we actually look at this problem from underneath so what's actually going on here is that if you actually work out if you actually solve this equation for just a equals 2 effectively what's going on is that you're actually looking at the equation expanded into the into the third dimension here and you actually get a shape that looks like this so he's actually worked out this this is actually represent a parabola it's actually a quadratic equation so the people that worked it out in actual fact worked out that these two solutions that were found independently actually have a wormhole connecting the two which involves a quadratic equation and so what they're able to do then is they're able to actually link every single answer by drawing parallel lines so this comes up here and in actual fact intersects is here and this goes off and shoots off and hits the graph again and intersects it again another time but at a different number that we didn't find before it's actually 8 and 30 and so you can actually do what's called vieta jumping you basically jump between these two curves at right angles and you find a secret super secret hidden infinite column of solutions and basically for any any solution here he basically worked out the connection between all solutions for a given whole number so this is a curve this is a curve yeah and you just draw sort of yes that's right and they're all they're all connected by parabolas pointing pointing at right angles underneath and they're going off at right angles so it's it's really I think one of the only times that research mathematics or something meaningful has come out of a maths problem from an Olympics because the error jumping is now something that you get taught I mean this is a game-changer really it's like a pros be flop in high jump but it's amazing because it's such a beautiful solution so the West German person who submitted the problem didn't know about fear to jump in no he did but the guy who solved it amazingly used v8o jumping in just this sleight of hand and solved it in a paragraph we've watched for this long so let me give you a little PostScript we mentioned that Terry tell the future Fields medalist only scored one out of seven on this notorious question six from 1988 that of course is with a caveat he was attempting the question the day before his thirteenth birthday and he still went on to become the competition's youngest ever gold medalist and in fact only 11 out of the 268 contestants at the Olympiad managed to completely conquer this question one of them was actually another future fields medalist no Bao Chau who actually got 42 out of 42 for the whole exam but a third future Fields medalist in the room a LAN de Linden Strauss also scored only one point on question six on his way to a bronze medal the boy who we mentioned receiving a special prize for his ingenious solution is right here Emmanuel Atanasoff from Bulgaria he picked up an overall silver medal but do you know who else got the full seven points for question six that girl there that's very Lena Stan Cova also from Bulgaria who won a silver medal in the Olympiad and if that name rings a bell it's because as vada Lena was Wester as we call her is a regular contributor here on numberphile and if you like you can go right now and watch some more of her videos it comes from my own textbook which was associated with the mass circle in Bulgaria it is called mathematical Olympiads and it was published in 1985 though the actual problem is the following right I'm able to I'm able to show you how beautiful how simple it is now to interpret this problem this big circle here Brady this big circle here what does it invert - well first of all it passes through the centre of inversion

Channel: Numberphile2

Views: 1,666,448

Rating: 4.875392 out of 5

Keywords: mathematical olympiad, vieta jumping, IMO, simon pampena

Id: L0Vj_7Y2-xY

Length: 16min 3sec (963 seconds)

Published: Tue Aug 16 2016