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
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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.
James Grime is the GOAT..
Nice episode! Thanks, Brady! It's great to see Dr Grime again! And I love the alfalfa joke! Took him a little bit to grasp it though 😁
I happily posted a comment on YT but it got lost among the almost 1k. Also, every time I posted it, YT would delete it because it contained links...
Links to a REPL I wrote! I got so hyped about the approximation thing that I went on and coded it myself, in Python. Only to get to a different result.
Dr Grime states alpha to be
109.1885...but I got to109.1883...Here's the link to the REPL: https://replit.com/@scorphus/TheGrazingGoat which you can run and see the approximation honing in on
109.1883...Click Show files to read the contents of main.py, where the code lies.I thought I had done something wrong, but the value of
rmatches,1.1587284730181215🤷🏻♂️It was a great exercise and made me enjoy the subject even more 😊