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,
Numberphile2. All will be revealed there. In the meantime, our thanks to the supporter of this video, the Great Courses Plus.
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Amazing secret solution to solving this problem
is actually a beautiful piece of observation.
It is mwah!