After centuries, a seemingly simple math problem gets an exact solution | Hacker News Reader

After centuries, a seemingly simple math problem gets an exact solution

quantamagazine.org

305 pointstechlover14159121 comments

The paper this article is about:

* A Closed-Form Solution to the Geometric Goat Problem, by Ingo Ullisch in The Mathematical Intelligencer. https://doi.org/10.1007/s00283-020-09966-0

The illustration in the article (I mean the top half of https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/12/Grazin...) was really helpful to understand what the question is. I know Quanta Magazine articles sometimes get some negative comments here on HN from readers who expect a different kind of exposition than what is suitable for a magazine of that sort, but for my part I'm really grateful to Quanta Magazine for bringing attention to papers like this, and writing nice articles about them with history, good illustrations, quotes from other mathematicians, etc.

For the impatient, Wikipedia has a short article on the problem and earlier progress: https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi... From this it's more clear in what sense this is progress: whereas earlier the answer r=1.1587284730181215178... (https://oeis.org/A133731) was known via more than one formula of the form r = (some function of r), only now do we have a closed-form expression of the form r = (some expression not involving r).

It's fascinating how a problem that looks at first glance like it belongs as a basic homework problem in a calculus textbook turns out to be so difficult.

I'm curious if there are lists of other problems that are similarly easy to understand in a few seconds, that seem like they'd be similarly easy to solve, but which turn out to be fiendishly hard like this one?

Especially ones that can be visualized easily geometrically like this one.

>I'm curious if there are lists of other problems that are similarly easy to understand in a few seconds

I too would love such a list. Can anyone point me if one does exist?

I'm struggling to come up with examples.

Maybe the 3n+1 problem? It's definitely easy to understand in a few seconds, and definitely fiendishly hard. I'm not sure if it seems easy to solve though :)

Another one that came to mind is about efficiently computing the permanent of a matrix[1]. Maybe understandable in seconds if you know how determinants work.

[1] https://en.wikipedia.org/wiki/Computing_the_permanent

3n + 1 is interesting because it seems so tantalizingly easy and beautiful at first glance. Its like looking at a stream where you can clearly see the stony bed and it appears to be ankle deep but once you take a step you realize the lighting has fooled you and you fall head over heels into icy waters.

It's so much more interesting if you happen to know about the "hydra game" (the Goodstein theorem), which is also a question about sequences and their termination at 0.

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

But this is easily proved (using infinite ordinals, but it seems it could be proved just by coming up with a similar concept like the infinite ordinal arithmetic, basically providing an upper bound for each step of the algorithm and showing that there's an eventual maximum to these and then there's a monotonic decrease, and that the number of steps are always finite).

"I have discovered a truly remarkable proof of this theorem which this margin is too small to contain."

I can't formulate proper google search query, but I once read funny story where spies were planting a papers with a math problem like it written by a kid, very simple looking, but the answer had actually crazy numbers.

Mathoverflow has their "long-open problems which anyone can understand". This includes things like the integer brick problem: Is there a brick where all its dimensions (width, height, breadth, face diagonals and main diagonal) are integers? And Singmaster's conjecture: How many times can a number (other than 1) appear in Pascal's triangle?

https://mathoverflow.net/questions/100265/not-especially-fam...

This led me to https://mathoverflow.net/a/38319 which has wonderful pictures!

One of my favorites:

If an ideal observer looks at the end of an ideal infinite cylinder, which is not deformed by perspective, when the cylinder is pointing directly in his line of sight he will see just a circle, but if the cylinder deviates slightly what will he see? A cylinder with finite length in his field of view? Or a cylinder that goes infinitely out of his field of view?

Is there a source for that one? because it seems like the whole problem is predicated on the phrase "not deformed by perspective".

I don't remember the source, the cylinder just helped to visualize the problem, probably you can replace the cylinder with a line. The idea is that you can see its whole length.

This doesn't seem to belong in the category of "problems anyone can understand". What do you mean by looking at a cylinder "which is not deformed by perspective"?

The only way I can think of to interpret this is that you pick a plane and project the cylinder onto it. (That's how you get the circle). But that's easy to do. Failing that, this looks like a "puzzle" that's supposed to sound interesting without actually meaning anything.

"which is not deformed by perspective" = parallel projection, as opposed to a conic projection. The further from the closest cap on the cylinder its apparent size remaining the same as the cap. In a deformed projection the opposite cap in infinity would collapse to a point. And that would defeat the thought experiment.

> And that would defeat the thought experiment.

What is the thought experiment? It's easy to know what the coordinates of the cylinder are. It's easy to project them however you want. Doing this doesn't tell you much. For the actual problem, we also have to ask:

- Is the rest of the world seen normally, or is it "not deformed by perspective" either? If there's no perspective for anything, most of what you see will just be blank white, the effect of the many different representations of different distant objects all overlapping one another in your vision.

- How are you "looking at" this cylinder? You're not using your eyes.

One of my favorites is Bellman's Lost in the Forest Problem. It is a 2D geometric problem that is easy to state, understand, visualize and draw. The escape path is unconstrained, so there must often be a series of rules and decisions to be made. Some doodling quickly reveals its subtlety.

It is also nicely phrased as a class of problems, because the forest's size and shape are known to the victim, but there are no constraints on what the shape might be. Some of the classes are solved, so you can chase down the spoiler solutions, but others are still open.

https://en.wikipedia.org/wiki/Bellman%27s_lost_in_a_forest_p...

Warning: don't read these unless you want your next weekend to disappear:

http://wardsattic.com/joomla/Download/BellmanForestProblem.p...

https://www.maa.org/sites/default/files/pdf/upload_library/2...

P.S. Previously on HN:

https://news.ycombinator.com/item?id=18001449

P.P.S. I have now added PDF links to the Wikipedia article.

There's always Fermat's Last Theorem.

There’s a lot of simple geometric problems like this that don’t have known closed form solutions. A lot of simple geometric time to intersection problems (comes up a lot in game physics) don’t have known explicit solutions for example. Finding them is interesting but once you have something you can use an iterative algorithm on to find a solution at arbitrary precision the pressure/incentive to solve them is reduced quite a lot.

Generally mathematicians would consider the number that answers this problem to be “known” in the colloquial sense, even before this better form.

Kepler's equation is a bit like this [1]. A body is moving in a known elliptical orbit, and you would like to know where it is at any given time - eg a quarter of the way through its orbit (so going from 'mean anomaly' to 'eccentric anomaly'). The starting point is a handful of simple equations, of motion and gravitation. But there is no analytical expression that answers the question - you have to solve it numerically, or approximate it with a sufficiently accurate Taylor series.

https://en.wikipedia.org/wiki/Kepler's_equation

One of my favorite classic examples is... a pendulum!

Typically we view pendulums as “simple harmonic oscillators” - basically equivalent to a mass on a spring, only going left-to-right rather than up-to-down.

But this is only an approximation. The differential equation for a simple harmonic oscillator is

x’’ = ax

which has a simple solution - sum of a sine and a cosine, aka a wave. (Here x’’ = second derivative of x with respect to time, or acceleration)

But the equation of motion for a real pendulum (from a free body diagram) is much more complicated and has no exact solution:

x’’ = sin(x)

It’s a very strange function whose second derivative is equal to the sine of the original function, and can’t be expressed in ordinary means. Any high school physics student can derive this equation of motion from Newton’s laws, but you need the assumption sin(x) ~ x to make any progress finding a solution.

I remember a take-home physics test in high school where there was a pendulum problem and I spent a full day trying to prove that x’’= ax is generally met by a trigonometric function. I eventually gave up, and was very surprised when I got full marks for saying to simply assume it (though the rest of the test suffered by my being stuck for so long). Besides the exponential formulation of sin and cos, I’m honestly not sure if that’s a unique answer to this day. Does anyone know?

There is an exact solution for the pendulum problem: https://www.researchgate.net/publication/39575612_Exact_solu...

I haven't studied it well enough to say more than that about it.

Thanks - I wasn’t actually aware of this but I meant to write “elementary” rather than “exact” so I’m glad you brought this up. I had supposed there was probably some incantation of special functions that solved the equation :)

It’s been a long time since I’ve studied ODEs and mathematical physics but this solution seems easier than I had guessed (in the sense that it’s accessible to advanced undergrads instead of specialists).

Weird paper. Eq. 32 is supposed to be an approximation to eq. 31, but they are the same equation.

Nope, that's pretty much all there is.

The space of all differentiable linear functions is a vector space, and the derivative is a linear operator on that space (so functions and derivatives follow most of the same rules as vectors and matrices). If D is the derivative operator and x is a function of t, the equation x''=x can be written as D^2 * x = x, or 0 = (D^2 - I)x = (D + I)(D - I)x. The dimension of the kernel of (D - I) is 1, and so is the dimension of the kernel of (D + I). So the space of solutions to the original problem has dimension 0, 1, or 2. But you can find two solutions, namely cos(i * t) and sin(i * t). Those solutions are linearly independent, so they span 2 dimensions. So all solutions are going to be of the form A * cos(i * t) + B * sin(i * t). The case for x''=ax is similar.

I agree with the main idea of your argument, but how can it be set in the "space of all differentiable linear functions" when trigonometric functions aren't linear? I think the argument would hold on the space of real analytic functions. Maybe that is more restrictive than necessary.

Ah yes that was a slip up. You're right, it should read the "space of all differentiable functions," or maybe "all complex differentiable functions" for extra precision.

Put a person each at the corner of a square, each facing the person in the corner in the clockwise direction. Have each walk at the same time at the same speed toward the person they're facing, so they all keep adjusting their direction as the person in front moves.

Describe the shape of the spiral path they trace.

The numbers pi and e are irrational. Is pi*e rational? How about pi+e ?

Most Diophantine equations. Someone has already mentioned FLT above. Problems of this sort is almost one of the defining characteristics of number theory.

>at first glance like it belongs as a basic homework problem

The culprit is that (most) education involves churning through problems and their often formulaic solutions, merely with increasing levels of difficult. You get used to simple problems having simple solutions by exposure. It would have been nice to understand as a child that there's so much simple stuff not figured out, in many areas.

But presenting open-ended problems in children textbooks sounds like a quick way to confuse and anger parents.

I first misread "goat tied to the inside of the fence" as meaning tied to a post in the center of the circle, and was confused at why everyone thought this question was so hard.

I was just drawing what you said on the whiteboard, and I was like.. uh, am I missing something or am I a super genius? I am positive the latter is incorrect.

I had to re-read the first paragraph multiple times until I understood the goat was not tethered to the center.

There's a picture of the problem like halfway down the article.

Yeah but I wanted to attempt to think about the problem myself before reading more on the article in case they gave away a very simple clue.

I initially assumed that the goat was tied to the fence in such a way that the non-goat rope-end could move completely around the circumference of the fence and not be fixed to a single point.

I was also at first confused, and I find the problem as published in 1894 (i.e. almost 130 years ago) much clearer:

"A circle containing one acre is cut by another whose centre is on the circumference of the given circle, and the area common to both is one-half acre. Find that radius of the cutting circle."

I think that formulation had to be at least mentioned near the beginning of the text, when not used in the first sentence.

Your goat eats the "donut" and GP's goat eats the "hole", each eating half of the area. Not a blade of grass was wasted; how satisfying!

> Your goat eats the "donut" and GP's goat eats the "hole", each eating half of the area.

Assuming the two goats are on equally long leashes, this doesn't come close to being true. They each eat circular arcs which run through half a radius. But the outside goat's circles are huge and the inside goat's circles are tiny; the inside goat covers 25% of the area.

I feel like "closed form" becomes a lot less special when you realize that things as simple as sin, cos, log, or even just sqrt don't have "closed forms" in the sense of "able to be expressed in terms of 'simpler' functions".

Indeed, there's always a convention as to what the building blocks are. Like chemists don't look for how the quarks are configured -- they are satisfied to know how the atoms are arranged. In some cases, finding out what the building blocks are is an interesting problem in itself, for instance what things are computable with geometric construction.

> chemists don’t look for how the quarks are configured

Neutron magnetic moment results from quark configuration, in terms of where the up and down quarks like to orbit each other within this gigantic neutron hadron thing.

This is useful for understanding the magnetic properties of materials up to a hundred angstroms thick! Fire cold neutrons into your multi-angstrom Great Wall and see how they scatter.

Oh it is special precisely because of that! We can almost always get the answer with numerical methods. Yet, numerical solutions are good almost only for certain values of parameters.

Analytic solutions on the other hand provide more insight. It can be differentiated, integrated, compared and maybe (big maybe) transform to some other/more familiar form. Finally it can be connected to whole body of mathematical knowledge. Analytic solutions are very satisfying for that reasons.

Closed solutions are small, very narrow subset of analytic ones, that’s why they feel special. But I suppose they become special only after some time spent with problems where closed form solutions are rare and unexpected. I suppose high school/college mathematics makes us way too used to rare special cases, such as closed form solutions.

> Unfortunately, there’s a catch. Ullisch’s solution ... can’t tell you, in a practical sense, how long to make the goat’s leash. Approximations are still required...

Ok, then in what sense is this an exact solution?

In the sense that sqrt(2) can be an exact solution, even though you can't write it down as a decimal.

No, sqrt(2) is exact and no approximation is needed -- it's the diagnoal of a 1x1 square. Goats and grass don't care about decimal number systems.

As the article says, the solution to the goat problem isn't practically constructible:

"It’s a bit more abstruse — the ratio of two so-called contour integral expressions, with numerous trigonometric terms thrown into the mix"

>No, sqrt(2) is exact and no approximation is needed -- it's the diagonal of a 1x1 square.

That's like saying that the goat-grass system as described is exact, and no approximation is needed. I can write 1 x sqrt(2) just as easily as I can write 1 x (goat-grass constant). We arbitrarily choose what constants and symbols are allowed when we use the phrase "exact solution." Philosophically, every solution is at most breaking down an answer into other solutions.

Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.

But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge.

Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.

But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.

You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.

A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)

> one that happens to be able to construct the goat-grass area exactly.

How? How did you get a string of the correct length?

Note that the tool you described is a compass.

> compass and straightedge

But those are not symbols. We're talking symbols (digits).

Who are we? The problem is to create a leash of a certain length. If you can make a leash exactly one unit long, then you can also make a leash exactly √2 units long; the fact that the ratio of the two numbers is irrational doesn't impede this in any way.

To remind, the context is:

> you can't write it down as a decimal

No, the context is whatshisface claiming that there is a (resolvable) tension between the concept of an exact representation and an infinite decimal expansion, and gowld responding that decimal expansions are irrelevant. That doesn't support the idea that we're talking about digital representation; it undermines it.

It's in closed form

It's clearer from the Wikipedia article (https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi...): earlier we only had expressions for r like https://wikimedia.org/api/rest_v1/media/math/render/svg/9758... and https://wikimedia.org/api/rest_v1/media/math/render/svg/a634... (from which of course one could numerically compute the answer r=1.1587…), but now thanks to this paper we have the expression https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... that does not involve r on the right-hand side.

Hm, if it's based on this ratio of contour integrals, shouldn't it be possible to do better than this? Like why would it be so hard to find the residues for these poles? Shouldn't that be just a bit of formal Laurent series manipulation? What am I missing here?

If the series don’t cancel out nicely (likely, I would guess), wouldn’t you end up with some infinite sum?

If so, as this example shows (contour integrals in a closed form?) “closed form” is loosely defined, but I think most would say something with an infinite sum wouldn’t be one (but then, https://en.wikipedia.org/wiki/Closed-form_expression#Analyti... says:

“In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.[citation needed]”

No, you shouldn't end up with an infinite sum; if you only want to know the series finitely far out, you only need to know finitely many terms.

However, it seems the problem is hard for other reasons. I had assumed, without checking, that they'd put the center of the circle at 3pi/8 because it's a pole. Nope! As best I can tell from some graphing tools, there is indeed precisely one pole in the circle, and it's on the real line, but it's close to the right endpoint; I don't know that it has any nice form. So I imagine that getting any sort of exact series expansion around there -- or even just getting exactly the first few terms, i.e. the first few derivatives there, which would be all you'd need -- would be difficult for that reason.

(Although, the higher the order of the zero, the more initial terms you'd need...)

Edit: Actually, I guess it looks like a zero of order 1? Except that doesn't make sense, because then the top contour integral would be zero...

Ha if only the article about mathematical equations had actually included any mathematical equations.

This is like those articles that are about a picture but don't include the picture.

Imagine an exact solution like this:

sin(cos(log(98234/123)+tan(exp(pi/5))-1)+pi + integral(complicated function(x) dx from 0 to 1)

It doesn't tell you in a practical sense how long to make the goat's leash. So you find the numerical version to however many decimals you want to get a useful approximation.

The unknown is isolated:

r=very-difficult-to-compute-expression

The word exact here is an inexact description of what sort of solution this really is - it's a closed form explicit solution. A closed form solution means that the equation is limited to to certain common mathematical operations and is finite in length. An explicit solution means that the quantity we are solving for is isolated on one side of the equation.

Since when do we call a function of a ratio of two integrals a closed form solution?

https://en.wikipedia.org/wiki/Goat_problem?Closed-form_solut...

https://en.wikipedia.org/wiki/Closed-form_expression?Compari...

Yes I would describe that as an explicit equation rather than implicit.

From TFA: * Of course, it won’t upend textbooks or revolutionize math research, Ullisch concedes, because this problem is an isolated one. “It’s not connected to other problems or embedded within a mathematical theory.”*

A island peak hinting at a submerged continent of mathematics.

Unfortunately since our brains evolved (under a regime of calorie cost vs survival benefit) and are therefore limited, we might never discover the continent.

Is this in any way to the problem of quadrature of the lune? https://en.wikipedia.org/wiki/Lune_of_Hippocrates

Hey Terence Tao, you might be a great mathematician but Ingo Ullisch is the GOAT!

Cool. Now solve it for a circular fence on the surface of a sphere. In fact, solve all four cases {exterior, interior of sphere} X {exterior, interior of fence}. Spherical trig can only make the solution(s) even more heroic, right?

Why not try hyperbolic space too?

You forgot about the hypergoats.

Surface of a sphere depends on how large the fence is compared to the sphere. If the fence is small, the answer is the same as the plane version. If the fence is as large as possible (an equator) then the rope needs to be precisely one quadrant in length, equalling the "radius" of the fence. If the fence encloses more than half the sphere... well, if it encloses all of the sphere (that is, it is a small circle with the "inside" declared to be the outside) then the rope is again one quadrant, so half the "radius" of the fence.

More interesting is a space-goat tethered to the interior of a hollow sphere, hypersphere etc.; no closed-form solutions for higher-dimensional cases, but the answer tends to sqrt(2) as the number of dimensions approaches infinity.

"[Fraser] worked out how long a rope is needed to allow a goat to graze in exactly half the volume of an n-dimensional sphere as n goes to infinity."

That sounds way too much like the punchline of a stereotypical mathematician joke.

This is a very interesting problem. One thing I'd be interested in researching is the ratio between the 3 arcs in the scenario (https://en.m.wikipedia.org/wiki/File:Goat_problem_2D.svg), I'm sure this is a mathematical rabbit hole that would be exhausting to dive into.

Not really an exact or closed form solution since it contains an integral. It is an explicit solution compared to the implicit one we had before.

> Not really an exact [...] solution

But it is exact.

> Not really [a] closed form solution

It's my understanding that what is and isn't "closed form" is rather arbitrary. Functions which are used frequently – like exp() – are elevated to closed form status, and yet you can't evaluate exp() in a finite number of steps. So how is the explicit solution to the goat problem objectively different?

Yes I also feel that way to some degree, but I just never considered an integral to be closed form. There is some argument to be made for exp, as it is considered an 'elementary function'. I was going off this statement from wikipedia on closed form expressions

"It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration."

Edit: I think if you say an integral is closed form, you must also admit that a limit is closed form, since an integral is defined in terms of limits (though technically more restrictive). In that case, you should also admit that we already had a closed form expression for this number, as it could be expressed as a limit of an iterative process.

> There is some argument to be made for exp, as it is considered an 'elementary function'.

But exp() is defined as the limit over an infinite sum, so why does it get to be an elementary function?

My point it that the distinction between closed form and non-closed form is arbitrary, and that there is no qualitative difference. In fact, limit and integral are just (higher-order) functions as well – and rather ubiquitous ones, so why aren't they considered elemental?

You're not wrong, in that closed-form is not a clearly-defined term.

However, mathematicians rarely do numerical calculations, so exp(), gamma, etc. are considered elementary functions, and there's no urgency to translate from symbols to numbers.

Physicists who do calculations might be more restrictive about what closed-form means if they intend to compute a numeric result.

https://en.wikipedia.org/wiki/Closed-form_expression

Source: I studied pure mathematics, and we never used calculators - it was laughable, in fact. Most of our series never converged in the first place. :)

I'm not a science student so sorry if this sounds dumb, but can you not just do a computer simulation and run it (bruteforce) with varying rope lengths until you get the right one for one acre, then repeat the process for other areas till you get a list of areas and needed rope lengths, then do some regression on that list to get a close enough formula?

That’s essentially what this replaces - which had been done without the aid of simulations.

You just need to do the simulation once. It's an elementary result that if the solution for a circle of radius 1 is k, the solution for a circle of radius 2 will be 2k.

That was already done in the 19th century without computers. But like you say, that gives only a "close enough" formula. Close enough for farmers, or even rocket engineers who might need much more precision, but not good enough for mathematicians.

Why do I have to go to Wikipedia to see the solution?

Would it have been that hard to stick https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... in the article?

From a journalistic standpoint, including the formula would require a longer digression into contour integrals than the one clause the article currently contains.

It’s also not clear what license that image is available under.

> It’s also not clear what license that image is available under.

A plain image of mathematical equation is not copyrightable, it's literally the MathJax output of a LaTeX equation ({\displaystyle r=2\cos \left({\frac {1}{2}}{\frac {\oint _{|z-3\pi /8|=\pi /4}z/(\sin z-z\cos z-\pi /2)\,dz}{\oint _{|z-3\pi /8|=\pi /4}1/(\sin z-z\cos z-\pi /2)\,dz}}\right)}). It does not have any copyrightable artistic design. And to the nitpickers - the pixmap output of a font is also not copyrightable under U.S. copyright laws. Even if it is, Computer Modern is available under a free license. And even if it's not, pure facts - such as math formulas - are not copyrightable, it would be trivial to write down the identical equation using another program and font.

Copyright is not an inevitable, divine, or natural right, it is only an utilitarian tool adopted by the Constitution and lawmakers "to promote the Progress of Science and useful Arts." Thus, there always exist things that cannot be copyrighted. It's also why fair use of copyrighted works is conditionally allowed (not relevant to this case). The tendency of people to assume that every single piece of data is automatically controlled exclusively under copyright is frustrating.

> Content is available under CC BY-SA 3.0 unless otherwise noted

Just including the actual solution in an article about this problem would be IMO a much better choice.

Probably because including equations dissuades the average person from reading a book or article

I wonder if someone comes up with a neat formula for the perimeter of an ellipse. Keppler gave up...

There are neat formulas, but those are just approximations, as an exact (closed form) is provably not possible.

https://www.reddit.com/r/math/comments/8gw9on/is_it_possible...

https://www.youtube.com/watch?v=5nW3nJhBHL0

This is probably written by people who don't have real-life experience with goats I guess.

I cannot even concentrate on the problem since I keep coming back to what the goat will do:

- jump the fence

- chew the rope

- probably more

In such cases, it's best to assume it's a special zero dimensional point goat, which only chews grass.

...in a vacuum.

How do you tie a zero dimensional point goat? How do you know he's in the fence at all? How does he chew the grass? Where does the grass go?

And having discovered the goat, how do you put a tether on something with no size?

You'd need a special rope to tie it as well, in addition to low-dimensional fencing and specialized grass.

Literally every one of these questions pertains to real goats, too.

... are you serious? You use an infinitely thin rope of course!

Slightly more seriously, you take the limit of a alpha-sized goat tied with an alpha-sized rope as alpha goes to zero from above.

Q: How do you tie a zero dimensional point goat? A: With a line.

Q: How do you know he's in the fence at all? A: From definition.

Q: How does he chew the grass? A: Using axioms.

Q: Where does the grass go? A: We need more grants^Wresearch to answer that.

First, imagine a spherical cow!

When I was a small child my parents took me to a petting zoo inside an amusement park. While an employee watched, the goat I had just fed started eating the shirt I was wearing. I was terrified. My parents, not wanting to hurt the goat looked to the employee for help. The employee said, "Yeah, they'll do that" and turned around, not concerned for me or for the goat. My parents eventually extricated my shirt and all was well. I have no idea why I was never afraid of goats (especially since I was afraid of trees for a period of about six months).

Yeah, goats will eat everything.

  Two goats chew film tape. One says:
  - This film is nice.
  - The book was better.

Are you a Scandinavian or is this joke international? (Haven't heard it in 20 years though :)

I'm from Poland. I've also heard it over 20 years ago.

The original statement of the problem talks about a horse. I wonder when it changed into a goat.

The better question is what farmer has a circular fence and is actively tying livestock to said fence. The fence should be good enough!

FTA: But in the 1960s, for mysterious reasons, goats started displacing horses in the grazing-problem literature — this despite the fact that goats, according to the mathematician Marshall Fraser, may be “too independent to submit to tethering.”

Plaudits to Steve Nadis for pulling every goat joke known to man for this article.

As a Naval Academy grad, it still took me a second to figure out why it seemed to be a thing at the Naval Academy... But then Bill the Goat. FIN. Beat Army.

Just use Goat Simulator: https://en.wikipedia.org/wiki/Goat_Simulator :-)

Finally! This was getting to be a real problem in my grazing goat as a service business.

Goats as a Service has been around longer than The Cloud. I think you already missed that gold rush.

You laugh, but growing up, my family used goats to keep brush down on our property, and would occasionally loan them to neighbors for the same.

https://www.telegraph.co.uk/technology/google/5297097/Google...

I think that is an actual service provided in Norway.

You are correct. I remember seeing some goats at Oscarsborg Fortress and I was told that they were there on loan or on hire. Looked it up now and it checks out.

The articles in the two links below affectionately refer to the featured goats as the "coast goat commando", which I think is just lovely :)

https://www.oblad.no/badebyen/kystgeitkommandoen-til-oscarsb...

https://www.oblad.no/badebyen/umb-geitene-pa-sommerjobb/s/2-...

Both of these articles are in Norwegian but basically they talk about the importance of keeping the vegetation in check, and that the goats are great at this, as well as the social value that goats provide to visitors. The goats featured here are owned by a University and were rented out to the Oscarsborg Fortress.

There are pictures of the goats also in the articles.

PS: For anyone not familiar with Oscarsborg check out the following links for some info and pictures:

https://en.wikipedia.org/wiki/Oscarsborg_Fortress

https://www.forsvarsbygg.no/no/festningene/finn-din-festning...

https://en.wikipedia.org/wiki/Battle_of_Drøbak_Sound

Using the defence batteries at Oscarsborg Fortress, the Norwegian coastal defence successfully sank the German heavy cruiser Blücher on 9 April 1940, forcing the German fleet to fall back.

https://en.wikipedia.org/wiki/German_cruiser_Blücher

California too: https://rentagoat.com

Are you referring to Goat-2-Meeting? https://www.sweetfarm.org/goat-2-meeting

Intuitively the formula should have the length of the rope, radius of the fence, and PI. And intuitively a rope length of the radius would cover half the circle, but a quick test of putting two goats in there shows they can't eat all the area. So the formula probably have a minus in it. Next I would try to figure out how large the goat circle outside the fence is because then i would also know the area inside...

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