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The Legend of Question Six - Numberphile

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Aug 16, 2016

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The Legend of Question Six - Numberphile
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  • Have you heard of the legend of question six? It's the question that actually stumped a Fields medalist.
  • This is one of the hardest problems
  • ever!
  • Well, it's something legendary.
  • I myself, when I tried to attempt this problem, it took me a year to solve this problem.
  • And when I finally solved it, I'm not, I'm not ashamed to say, I cried. To put question six into context, alright,
  • I'm talking about an international
  • mathematical Olympiad
  • problem. The international mathematical Olympiads is the premier event
  • for young people who are really good at math, they haven't gone to Uni yet,
  • they're under twenty, and it's for young people to go and compete on the international stage, right? And they're fighting for
  • ultimate mathematical glory.
  • Announcer: "A team consists of six kids from each country.
  • "These 'mathletes' are required to answer six questions worth seven points each."
  • So just like the Olympics,
  • they're held in a different country every year. They don't do it every four years, we need the competition. And we're talking about somewhere between
  • 220 to 260 students come from all different countries. And so when you think about it,
  • this is the best of the best. It's about being able to solve
  • awesomely hard problems.
  • But there's one epic problem.
  • Which means a lot to me, because it's from the maths Olympics in my home country of Australia.
  • So we held the math Olympics in 1988. And at that time, there was one particular problem.
  • It was the last problem of the whole event. Over two days,
  • there's three problems each, and on the second day the last problem was problem six.
  • And this one went down as one of the hardest ever. For a long time it wasn't surpassed.
  • Now, there's actually been harder problems that have come up recently. But for a long time, this one was just
  • amazingly difficult. And not only that, it was like, people who did solve it were, like, revered.
  • So for the 260 odd people that competed at this maths Olympics,
  • only eleven people solved it perfectly. And only one person solved it awesomely.
  • And, I have to say,
  • Terence Tao was competing there, and he only got one out of seven.
  • One mark out of a total of seven for this problem, which means he didn't get it right. And Terence Tao
  • went on to win the Fields Medal in 2006.
  • He's awesome, and he's Australian! Can I just say that please?
  • Brady: "And he's from Adelaide."
  • And he's from Adelaide, where I'm from as well, yeah.
  • Brady: "And me!"
  • And you, Brady.
  • I didn't know whether we could mention that.
  • Let's actually, let's actually cut, cut Terence Tao some slack.
  • The Fields medalist, the guy who's, like, coming close to solving the twin prime conjecture.
  • He's actually pretty awesome, because he was only thirteen years old.
  • Brady: "And that's young, is it?"
  • Well, he holds the record as the youngest person to ever have won a gold medal at an international math Olympics. So, it's pretty good.
  • But he still didn't at the last one right.
  • Announcer: "Terry Tao was a toddler when his parents noticed he had a gift.
  • "At a family gathering, they found their two-year-old giving older children a maths lesson."
  • I mean, these are problems that you don't find in a maths book.
  • There's no, you don't look in the back of the book for the answer for these problems.
  • These are really hard problems. And, and in a way, they're actually designed
  • to kind of throw you off if you know high school maths too well. So say, if you really know how to solve quadratic equations,
  • and you see something that looks like a quadratic equation, it almost like throws you down the wrong path.
  • It's almost like, nuh-uh, you're at the international maths Olympics.
  • Question six
  • was pretty special. It was come up by this
  • crazy hardcore West German guy, at a time when there was a West Germany.
  • And he submitted his question to the mathematicians who are in charge of selection.
  • They're given six hours to solve question six.
  • Brady: "These are like the experts, like test, test driving it. They're test driving."
  • Yeah, yeah, yeah, they're test driving it.
  • They're gonna see whether it's gonna make the cut, you know? Because, like, if it's too easy,
  • turf that, right?
  • Well guess what?
  • They couldn't solve it. After six hours.
  • Question six, after six hours,
  • still stood, yeah?
  • So you'd think, hey, this is too hard.
  • But what do they do?
  • They had the courage to submit it.
  • Brady: "They put it on the test."
  • They put it on the test. And in the original copy, they actually put a double asterisk next to it. Just to make sure
  • everyone knew this was hard.
  • Brady: "So don't waste your time on it."
  • Well, not necessarily don't waste your time,
  • just don't beat yourself up if you can't get it. And so here, I've actually got a copy, July 16, 1988.
  • So it's held over two days. This was the second day, and it was the third question on the second day,
  • so therefore, it's question six.
  • Now if you have a look here, the time to answer all three questions: four and a half hours.
  • So that means, on average, you had
  • ninety minutes to solve this problem.
  • And if you can't work out that it was ninety minutes,
  • I think you're gonna struggle with the whole exam.
  • Given the fact some of the best mathematicians in the world were given six hours, and they couldn't solve it, so
  • they gave students ninety minutes on average.
  • Maybe it could be more. Maybe you can finish these ones really quickly,
  • and then you just sit there sweating. And if you have a look at it,
  • it's the shortest one on the page. Look how tiny it is. It says, let a and b be
  • positive integers such that a times b plus 1
  • divides a squared plus b squared, show that a squared plus b squared
  • divided by a times b plus 1 is the square of an integer.
  • It's not even simple to explain. I mean they have actually covered up
  • part of what, what you need to understand. Okay, let me just write it out for you.
  • Okay, so. So what this says is that first of all, a and b
  • can be
  • elements, they can be 0 1 2 3, blah-blah-blah, they can be any whole number, including 0. Let's stick them into this equation:
  • a squared plus b squared
  • divided by a times b plus 1. Okay, so let's actually do a little bit of a chart for this.
  • So, possibly, it could equal a fraction. So if it does equal a fraction, the question says, no one cares.
  • Really, question six doesn't care.
  • But, if it just so happens that once you divide
  • a b plus 1 into that, and you get
  • a whole number, it then says, oh, that whole number will also be a square. It means that it will be
  • something to the power two.
  • Brady: "Like 25, or-"
  • Yeah. 25, 36, 49, 121.
  • It'll be some whole number multiplied by itself.
  • So only if it's a whole number, it actually will also be a square number.
  • So I suppose, I suppose one way of saying it is that
  • the only solutions to this
  • given whole numbers, when you stick in whole numbers, are fractions
  • and squares.
  • And the thing is, is like, why squares?
  • Perhaps we should find out.
  • So there's the history of the problem explained. And by now,
  • I hope you've cracked out your pencil and paper to try it. For those wanting the solution a bit sooner,
  • the next video will be posted on our second channel,
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  • is actually a beautiful piece of observation.
  • It is mwah!

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Simon Pampena discusses the famous Question 6 from the 1988 International Mathematical Olympiad. More links below...

Second part of this video: /watch?v=L0Vj_7Y2-xY

International Mathematical Olympiad (includes links to all previous Olympiad results and papers): https://www.imo-official.org

Simon Pampena: https://twitter.com/mathemaniac

Terence Tao: http://www.math.ucla.edu/~tao/ (thanks to him for the photos)

Support us on Patreon: http://www.patreon.com/numberphile

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