So I'm going to tell you about the goat problem, yeah? It's maybe- it's not the greatest of all
time but it's, you know, it's good enough, we'll do it, it's all right. This is kind of like a
school kind of problem but it's actually hard to solve. For a long time we did- we only had
an approximation of the answer, until recently. And we actually for the first time have an exact
answer to the goat problem. So let me tell you what it is - it does involve a goat. So
you've got like a fence going around a circular field. And you have tethered a goat to the fence,
so to the side of that circle. So this is some sort of rope of length - what do we call this?
Length r. So the field has a radius of 1 - why not - and now we're going to tether a goat to the side
of the field and let's say the rope has length r, okay, let's- there you go let's do a little goat.
(Brady: That is confusing that we made it r.) - Well yeah it's kind of r because if you look
at where the goat can reach it's kind of a radius of a circle. So this is where the goat can graze.
So this is a fixed point where you've tethered the goat; the goat can now roam as freely as that
tether will let him and so he can graze on the field. And they question in the goat problem is:
how long should the rope be if you want the goat to graze on half the field? And that feels like
something you might be able to solve maybe with school level maths - you can get quite far with
school level maths. Let's see how far we can get. - (It feels easy)
- Yeah, it feels easy. And this is why people are so frustrated when you tell them this
problem. I'll tell you what, I gather this is a problem that they used to teach a lot in U.S Navy
academies. I don't know if they're teaching this as an example of being used to difficult problems;
because they would teach this in these navy academies and they would say - and you can
do this - there's no exact answer. And people were frustrated and go,
surely this must have a exact answer, but no exact answer was found until recently, and
it involved some hard maths. - (How old is school student attack this problem?) So let's do another
circle. Let's make that the centre of the field - do a little x at the centre of the field - and
let's say my goat can eat some sort of circular area like this, something like that. So my goat
can reach all the way to the fence here, and my goat could reach all the way to the fence here,
and then he can eat all this kind of curve here. So it's a fraction of a circle with a radius of r.
This angle here that I've just made, let's give it a name, alpha. I'll use alpha for the angle, okay,
quite good, it's a good name for for an angle. - (Good grassy number; grassy,)
(alpha- alfalfa.) - You think alpha is grassy?
- (Alfalfa, yeah, it seems- I don't know.) So, this is how you would start to calculate
this with a bit of trick- I'm not going to go through all the details of the trick because
it's boring. So what is the area of the field? Well it's a circular field, it has a radius of 1,
so you use your traditional formula for working out the area of a circle. And the field is going
to have an area of pi. But we want the goat to eat half the field, so pi over 2.
Happy with that? - (Okay, pi over 2
is what the goat- we want the goat to have.) - Yeah, exactly. (Alright, now you've given us
his eating- well some of his eating area there.) - Yeah so this is some of his eating area
isn't it? I've kind of missed off some bits here, I'll come to that as well. So what I'm going to do
is a little bit of school level trig; so when I say trig I mean, you know, it's angles and
signs and cos's and triangles and things like that. The length of the rope is going to be
2cos and then it's alpha over 2, it's half of this angle here. But I've kind of- doesn't help
me work out the length of the rope, that's what I'm trying to work out, because now I've just
sort of change the problem because I need to work out what the angle is. Okay, so that's what we're
doing next. So that part is not so hard. Working out the angle is where it starts to get hard,
okay. So if you calculate this, what we can calculate is alpha is a fraction of a full
circle so I know that this area here is a fraction of a full circle of radius r. I can also work out
if I start at the centre here a different circle, because I've missed off these bits here, so let's
try and get those back. This is a fraction of a circle with a radius 1 and so we could talk about
this angle and we could work out- if I worked out at that angle it's going to be related to alpha
so I'm not going to do all those calculations. But we can work out this fraction of a circle as well
and then add it together and then we've got this area; but oh no wait, hang on, this diamond shape
that I've got here - can you see it's overlapped? So I've got an overlap, kind of like two triangles and
it's all overlapping. You remove the overlap and then you've got the area the goat has eaten. I've
not gone through the trig details because we would just be here looking at sins and cos's
for a long time. But this is what you would get: pi plus alpha - so that's our angle - cos of alpha minus
sin of alpha. That's the area that the goat is eating - hopefully you can see why I didn't
go through all the details because it's actually quite a nasty calculation to do. What we're going
to get then if we solve this is we're going to find the angle alpha. And then that's okay because
I have a formula for the length of the rope there, that's where we come- we fill- we do one last step
and then we'll get that length of the rope. So it's actually calculating that angle that's important
for us for now. And we want this area that the goat is eating to be half the field; and half
the field is pi over 2. So we set this equal to pi over 2. We're trying to solve this equation; it's
not a particularly nice equation, it's trig and in fact this is difficult to calculate because
it's called a transcendental equation. So it sounds like a nice name but it's a difficult
kind of equation to solve. A nice equation to solve is what we call a polynomial, and that's the kind
of things you get where you have x squared plus x minus 6 - those kind of things. They're quite easy
to solve and when you do you get an exact answer at the end so I can then say x is equal to - and I
could write something down. It might have square roots and horrible things in it but I can give
you an exact answer. For a transcendental equation I'm afraid there is no method for giving you an
exact answer; the only way to solve it - and it's the angle we're trying to solve - is approximations. It's
the only way to solve it. And so that has been done, so we have an approximate answer for that angle. So
we're getting closer and the approximate answer is it's about 109 degrees plus some decimals. It's 109
degrees point 1885- something something something. And we're not quite finished the goat problem because I want to
know the length of the rope. And then really the last step is the easy step, yes the easy bit
of trig showed us that the length of the rope should be 2 cos alpha over 2; and that's going to
be - and again it's only an approximate answer - but it is this answer: 1.15872847301...
(So just longer than the radius of the field?) - So it is- yeah it's a little bit longer than the radius
of the field. I would have expected- if you asked me to guess I would have gone um root 2, square
root of 2, something like that. That's what the answer expected it to be. I think that's where some people
get frustrated with this because they want it to be an answer like that; but it's not what we've
got. And there is - well until recently - there is no exact answer, it's that dot dot dot that shows you
it's not exact. It's a decimal, goes on forever, and yes. I mean we've got an equal sign, but equal dot
dot dot. We can make this more accurate, you know, you can make better approximations but I could
never tell you like an exact answer, like a little equation or a little formula telling you what the
length of this rope is. What's surprising about that problem, why it's surprisingly hard is because the
3-dimensional version of it is easier. The 3-dimensional version is a bird in a cage.
Imagine having a spherical cage of radius 1 and you've got a bird tied with a thread and then
that bird can then fly wherever it's allowed to fly limited by that thread. So it can fly around; what
does the length of the thread need to be to be half the volume of the sphere? That should be a
harder question - it turns out to be easier. Umm yeah you can actually solve it. Here's the answer: the
equation you get to solve is something like this, it's 3r to power 4 minus 8r cubed
plus 8 equals zero. I mean it's a traditional mathematical equation, it's a polynomial. And if
it's a polynomial that will have an exact answer. It might have square roots in it; it might look
a bit nasty but it will have an exact answer. In fact when this was solved - Marshall Fraser
was a mathematician who solved this - he did give an approximation for this answer. His
approximation was that thread was 1.228545 and then something something something.
So he did originally give an approximation. But a few years ago
um an exact answer was given and this is kind of nice for me, it was a couple of friends of
mine who did it, so this is nice I get to give them a shout-out. This is Nick and Graham Jameson;
actually Graham was my lecturer at university, and Nick is my friend for other reasons. So hey hey
guys, this is really nice. And they actually worked out the exact answer for the bird version of the
goat problem and it's this monster - all right, you ready? And it is a monster, so are you ready?
(I'm ready) - My friends actually broke it down into two parts because it's such a difficult thing to write
down. They define something called a and this is a horrible thing as well; so a is 16 over 9 plus
4 over 3 and inside a couple of brackets we've got 4 plus square root of 8 cube rooted
and then we add on another bracket here which is going to be 4 minus square root of 8 in a
bracket and we're going to cube root that as well. And then that's just halfway there, because then
their length of the rope thread for the bird was going to be 2/3 plus a half of square root of a,
that's where that a comes in, minus and then we've got another horrible thing, another square root
of 16 thirds minus a - wow - plus 128 over 27 and a square root of a down there as well. Whoa - it's
ridiculous isn't it? It's outrageous but it is an actual exact answer to this problem. - (So I could)
(write that down as a decimal number?) - You could write that down as a decimal number but- I mean
that's what Marshall Fraser did; he didn't want to calculate that outrageous expression so he said
well, I'll just approximate it and maybe that's kind of more useful in some situations. But in
this case, this is a solvable problem. Umm actually in higher dimensions it gets easier; if you keep
going up into the dimensions - 4 dimensions, 5 dimensions so on - the Goat Problem, the
length of the rope, starts to tend to an answer and it tends to 1.41421...
Oh it's the square root of 2. It's- it actually tends to the square root of 2, so it
actually gets easier somehow as you're going to higher and higher dimensions. And I would have
expected the answer to be square root of 2, just intuitively that would be my guess. Square
root of 2 feels right to me - maybe that's just me but it just feels right. But it's the 2-dimensional
one that caused all the problems. It's just really really hard to solve - uh well until recently, we now
have an exact answer. This has finally been solved, it's been solved by a German mathematician called
Ingo Ullisch; and to solve it he actually had to use some higher level maths. He used complex analysis
so he calculated- that uses complex numbers. And he calculated the areas using functions in complex
numbers, they were like sin functions and cos functions, but now they're using complex numbers
instead of the regular numbers. And worked out the area using this kind of maths and he found
an exact answer; he found that he found the exact answer for that angle alpha. Once you've got that
then you can stick it into the formula we had before. And it looks like this: so our angle alpha,
that's what we wanted to know. It's an integral, it's a complex integral, that's what that symbol
means. And we're going to evaluate this - when we do evaluate it - at z is a variable and oh what we've
got is going to be z minus 3 pi over 8 equals to pi over 4. This is where you evaluate it and
then this is the function; it's z divided by the sin of z minus z cos of z minus pi over 2. Put
that in a bracket and let's integrate that; that's the dz, that's how you do that kind of thing. And
we're dividing by another integral which is kind of similar - it's outrageous isn't it? But
it's an exact answer; I mean for the first time we've got alpha equals a formula. - (And how long's the rope?)
And how long's the rope? I mean I'm just going to go back to our approximation because it
was easier to work out, which was what I said earlier which was 1.158. - (So it gets pretty much the)
(same it's just now-) - Oh it is the same, it is the same. In fact I will tell you this, it's really nice
to have an exact answer - I mean a formula for the first time. Although this formula is hard to work
out and if you wanted to work it out you'd be better off approximating the values anyway. So
it's still calculated by approximations. But hey we've got the first formula for the angle alpha!
(So you would then put that into-) - You then take that alpha, put it straight into the formula that
we had with our trig. So the trig is still there uh as our final step. - (And that- and that's just)
(going to change the length by like a billionth of a nanometer sort of thing?) - I have no idea. Because I don't think
it's actually been calculated because it's hard.
(So we know the exact number but no one would probably bother calculating it. You couldn't put that into Wolfram Alpha?)
I don't think we can- even Wolfram Alpha I think would struggle to calculate this answer. - (Why did this mathematician do this? Why)
(did he spend his time just because-) - I don't want to speak for him but I can guess if you want? I mean
it is an exercise in his methods in mathematics. It's- it's practice. And then, you know, this is
not an important problem uh but it is um a way to practice, you know, your tools. And then you can use
that practice one day in the future for problems you do care about.
Okay, here's a puzzle for you:
it involves rotating a die, using the numbers on the face, and moving it around this board.
It's the latest brain bender from Jane Street, today's episode sponsor, and to check out the
details have a look at their website. There's a nice clear explanation there and you'll also find
a wall of fame for people who have already cracked the puzzle - maybe your name could be added. See the
link on screen and in the video description. And while you're there why not check out some of the
latest jobs and opportunities and things going on at Jane Street. They've got officers in the likes
of New York, Hong Kong, and London - a career at Jane Street could be just the thing for someone with a
puzzle solving mind like you. - ...every row column and the diagonals, I also want them to include
a club, a heart, a spade and a diamond. - It's like a Sudoku. - It is like a Sudoku! So try it out, it should
be the same kind of thing as a Sudoku. There is a solution I'm not trying to catch you out.