Grazing Goat Problems | Hacker News Reader

Grazing Goat Problems

quantamagazine.org

113 pointsnsoonhui28 comments

When I was about 14 (and in the advanced math set) my math teacher gave me this problem. Specifically the one with the goat in a square field tethered in the center of one side.

I spent ages trying to solve it and managed to convince myself that it didn't have an analytical solution - well at least not one I could come up with. So I invented an iterative process to binary search the solution on my calculator. I came up with an answer in the end to 5 or so significant figures which I was happy with.

That was the first math problem that I met that didn't have an analytical solution but I could solve so it was one of those breakthrough moments in my maths learning.

My math teacher was happy with the solution. She'd given it to us deliberately as an example of a problem with a solution, but no analytical solution as a kind of trick question in order to make us think. Great teacher!

As a keeper of goats, the answer is always “they eat whatever you don’t want them to eat”.

Goats and fences don’t exist in the same quantum state unless observed.

I imagine the answer would be a little harder to calculate if you had to introduce the statistical probability of the goat eating the rope.

That depends on if you want the goat to eat the rope or not. If not, 100%.

Something that gets my goat - goats don't graze, they browse.

Great article though!

Thanks for teaching me something new!

https://en.wikipedia.org/wiki/Grazing_(behaviour)

https://en.wikipedia.org/wiki/Browsing_(herbivory)

With the latter containing the wonderful line 'If the population of browsers grows too high, all of the browse that they can reach may be devoured.'

Suddenly the Moroccan argan tree goats make a lot more sense!

If your favourite browser the Moroccan argan tree goat too?

It’s a close second behind Firefox ;)

Based on the definitions I've been able to find, it would seem browsers are animals that mostly eat by browsing, and grazers are animals that mostly eat by grazing, but a browser can graze and a grazer can browse.

The goats in these problems are confined to fields of grass and that is all they can eat, so it seems they are in fact grazing goats (and probably unhappy goats since they would much rather be browsing).

Also, who the hell makes a circular barn?

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

More "real world" complications: how far does the rope stretch, and how far does the goat stretch?

I ran a goat for years on a 300ft runner cable, with a 20ft chain sliding freely on that. She worked out how to tie some incredible knots. I had to cut down two trees on the run because she figured out how to use them as fulcrums to lever the carriage link open.

Oh, for a second I thought this was about Goat Ops [0].

[0]: http://www.goatops.com/

Goats are the pentesters of the barnyard or as my step-dad says a goat will show you any weakpoints in your fence.

The article spends a long time building up the problem, but doesn't give the closed-form solution, which I find frustrating.

It's here on wikipedia: https://en.wikipedia.org/wiki/Goat_problem#Closed-form_solut...

Problem is not optimized by assuming spherical goats.

Nitpicking I know,

but "we now know exactly how long the rope must be to get an area of 50 square units"

does not work with a formula that contains pi and additionally a square root, does it?

We know exactly what pi is, it is pi. The circumference of a cirle is exactly 2rpi. In this way we know the exact answer. But knowing it is "pi" - not the number pi 3.1415926535... - we don't know how long to create a rope, for that we would need to know the "complete" number of pi.

(Great article though)

Actually, it's just as possible to get a rope of the exact length pi/2 as it is to get a rope of the exact length 1. Which is to say, it is strictly speaking impossible (heisenberg's uncertainty principle and all that), but you can get an approximation as good as you want, being far more limited by physics than by the properties of Pi.

Now, if you wanted a rope of length 1+i, that would be a different problem.

Edit to add: whether two objects can be exactly Pi meters apart is a different question, one with no known answer - is space infinitely subdivisible, or is there actually a minimum possible length? We certainly can't measure that distance, but whether it exists or not may be a different question (some physicists will claim that if it can't be measured it doesn't exist in principle anyway, others are more open to this idea).

In current QM space itself is assumed to be infinitely subdivisible/continuous, but there are theories like Loop Quantum Gravity where it's not, and interpretations of QM like Copenhagen where unmeasured quantities don't have any definite value.

We can get "exact" lengths in rational multiples of pi by cutting a rope/string to exactly one circumference of a circle of radius (length/2pi). Similarly you can get square roots by using the hypotenuse of the appropriate right triangle.

I don't think you can cut a String exactlyu to the circumference of a circle of a certain radius. It will always be longer or shorter.

Are you saying it's possible to cut a rope to length 3?

you could cut it to the point that it's effectively 3 from yours or mine point of view but mathematically its likely 3.0000000000000000000000000001 or something.

Precision on any cut in the real world will near impossible to do in absolute.

I think you need to clarify your definition of "exact" for this to be a productive line of discussion. Are you looking for a closed-form solution, a rational number, something that can be constructed with compass and straight edge, ...?

I wish i could think of interesting solutions to problems like this and others.

Is it involves Taylor series?

Goats do not graze though.

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