OT: Solved! Finally!! The Infamous Goat Problem!!!

[RegisteredUconn]

No, not the Dee/Maya/Stewie GOAT problem. The other -- lower case o-a-t, as in the animal -- Goat problem.

To be 100% honest, I wasn't aware of any goat-the-animal problem, infamous, famous, or otherwise, until I read the below link. And, as long as I'm being honest, I can't say I understand whatever they're claiming is a solution either. The only reason I'm bringing it up here is because there's a reference to Zeno's Paradox which I did solve many years ago but which still trips up many of the scientifically/mathematically minded today, and I'm wondering if others here have spent time solving it on their own.

Zeno's Paradox has been posited in many forms. I first began pondering it in the form of a hypothetical ball that, when released from a height (the actual distance isn't materially important), strikes a surface and rebounds to exactly one-half the height from which it was dropped. The ball then falls from that new height and rebounds to half again. This continues ad infinitum, the ball always bouncing exactly half as high. The question, and ensuing paradox, is how long will the ball bounce. Again, this is a hypothetical ball. There's no friction or heat dissipation or atmospheric resistance or uncertainty principal to confound things. Just pure mathematics. The quick answer, the obvious answer, and the wrong answer (even in the hypothetical world) is it won't, i.e., the ball will continue bouncing forever. Anybody know why the ball will stop? I learned the reason in about 6th grade arithmetic when I was introduced to decimals but didn't learn of Zeno and his paradox until 15 or so years later.

A Mathematician Has Finally Solved the Infamous Goat Problem

It only took a couple centuries.

finance.yahoo.com

 

[bluedevil]

The answer and 5 dollars will get you a mcrib...

 

[Bama fan]

No, not the Dee/Maya/Stewie GOAT problem. The other -- lower case o-a-t, as in the animal -- Goat problem.

To be 100% honest, I wasn't aware of any goat-the-animal problem, infamous, famous, or otherwise, until I read the below link. And, as long as I'm being honest, I can't say I understand whatever they're claiming is a solution either. The only reason I'm bringing it up here is because there's a reference to Zeno's Paradox which I did solve many years ago but which still trips up many of the scientifically/mathematically minded today, and I'm wondering if others here have spent time solving it on their own.

Zeno's Paradox has been posited in many forms. I first began pondering it in the form of a hypothetical ball that, when released from a height (the actual distance isn't materially important), strikes a surface and rebounds to exactly one-half the height from which it was dropped. The ball then falls from that new height and rebounds to half again. This continues ad infinitum, the ball always bouncing exactly half as high. The question, and ensuing paradox, is how long will the ball bounce. Again, this is a hypothetical ball. There's no friction or heat dissipation or atmospheric resistance or uncertainty principal to confound things. Just pure mathematics. The quick answer, the obvious answer, and the wrong answer (even in the hypothetical world) is it won't, i.e., the ball will continue bouncing forever. Anybody know why the ball will stop? I learned the reason in about 6th grade arithmetic when I was introduced to decimals but didn't learn of Zeno and his paradox until 15 or so years later.

A Mathematician Has Finally Solved the Infamous Goat Problem

It only took a couple centuries.

finance.yahoo.com

Quite a dichotomy you present there. Both Heisenberg and I are uncertain as to your claims. Old Zeno was fond of reductio ad absurdum; and Socrates, smart old dude that he was, thought the Z was the bomb.

 

[eebmg]

Ok. I will take a whack at the ball problem. First, consider total distance travelled which is slightly easier than time

Each successive height is 1/2 previous so sum of all heights is proportional to 1+1/2+1/4+1/8+...
=sum(1/(2^n)) which clearly converges to a finite value (Geometric Series)

Geometric series - Wikipedia

en.wikipedia.org

Now for time, the time to travel a height h under an acceleration (g) is t=sqrt(2*g*h) so each time is proportional to sqrt of each height so total time is proportional to 1+1/sqrt(2)+1/sqrt(4)+1/sqrt(8) + ... = sum(1/2^(n/2)). Now this sum is not a geometric series but the sum can be shown to converge to a finite value using the "root" test

Root test - Wikipedia

en.wikipedia.org

 

[temery]

No, not the Dee/Maya/Stewie GOAT problem. The other -- lower case o-a-t, as in the animal -- Goat problem.

To be 100% honest, I wasn't aware of any goat-the-animal problem, infamous, famous, or otherwise, until I read the below link. And, as long as I'm being honest, I can't say I understand whatever they're claiming is a solution either. The only reason I'm bringing it up here is because there's a reference to Zeno's Paradox which I did solve many years ago but which still trips up many of the scientifically/mathematically minded today, and I'm wondering if others here have spent time solving it on their own.

Zeno's Paradox has been posited in many forms. I first began pondering it in the form of a hypothetical ball that, when released from a height (the actual distance isn't materially important), strikes a surface and rebounds to exactly one-half the height from which it was dropped. The ball then falls from that new height and rebounds to half again. This continues ad infinitum, the ball always bouncing exactly half as high. The question, and ensuing paradox, is how long will the ball bounce. Again, this is a hypothetical ball. There's no friction or heat dissipation or atmospheric resistance or uncertainty principal to confound things. Just pure mathematics. The quick answer, the obvious answer, and the wrong answer (even in the hypothetical world) is it won't, i.e., the ball will continue bouncing forever. Anybody know why the ball will stop? I learned the reason in about 6th grade arithmetic when I was introduced to decimals but didn't learn of Zeno and his paradox until 15 or so years later.

A Mathematician Has Finally Solved the Infamous Goat Problem

It only took a couple centuries.

finance.yahoo.com

I would have thought they could have just gotten a goat and some rope and actually done it.

 

[RegisteredUconn]

I would have thought they could have just gotten a goat and some rope and actually done it.

I understand they actually did that but the goat would never divulge the answer.

 

[eebmg]

I would have thought they could have just gotten a goat and some rope and actually done it.

The goat would chew through the rope??

 

[RegisteredUconn]

Ok. I will take a whack at the ball problem. First, consider total distance travelled which is slightly easier than time

Each successive height is 1/2 previous so sum of all heights is proportional to 1+1/2+1/4+1/8+...
=sum(1/(2^n)) which clearly converges to a finite value (Geometric Series)

Geometric series - Wikipedia

en.wikipedia.org

Now for time, the time to travel a height h under an acceleration (g) is t=sqrt(2*g*h) so each time is proportional to sqrt of each height so total time is proportional to 1+1/sqrt(2)+1/sqrt(4)+1/sqrt(8) + ... = sum(1/2^(n/2)). Now this sum is not a geometric series but the sum can be shown to converge to a finite value using the "root" test

Root test - Wikipedia

en.wikipedia.org

A+

The trap lies in the erroneous conclusion that the sum of an infinite number of terms must itself be infinite. If you are like me you were introduced to repeating decimals (anybody want to calculate the decimal equivalent of one-third?) in elementary school. Calculus, of course, got all over "limits." As eebmg demonstrated even the hypothetical ball won't travel greater than two units. Since the distance is finite and the ball is always in motion, it will travel the distance.

 

[and one]

I think of accumulating knowledge as opening new rooms in your mental house. Read a little about astronomy and a new door opens although the contents of the room are dim. Read some more and the furniture becomes visible and read some more and the patterns in the upolstory come to light. Read enough and the new astronomy room becomes familiar and comfortable, albeit with nooks and crannies that have yet to be explored. Live a full and curious life and your mental house becomes a mansion.

I say this as a preface to admitting that my mental house has a door labeled math that remains only very slightly ajar. I've read a good deal of math stuff, like eebmg's proof, and find it interesting dispite not having a clue as to how the calculations are done. I like science; I've read alot on astronomy and physics and geology et. al. over the years but know that without the skills to understand the underlying math that I'm reading simplified translations. Which, at this stage of the game, is as good as it will get. But kudos to all the eebmg's of this world that have tamed the math beast!

 

[eebmg]

I think of accumulating knowledge as opening new rooms in your mental house. Read a little about astronomy and a new door opens although the contents of the room are dim. Read some more and the furniture becomes visible and read some more and the patterns in the upolstory come to light. Read enough and the new astronomy room becomes familiar and comfortable, albeit with nooks and crannies that have yet to be explored. Live a full and curious life and your mental house becomes a mansion.

I say this as a preface to admitting that my mental house has a door labeled math that remains only very slightly ajar. I've read a good deal of math stuff, like eebmg's proof, and find it interesting dispite not having a clue as to how the calculations are done. I like science; I've read alot on astronomy and physics and geology et. al. over the years but know that without the skills to understand the underlying math that I'm reading simplified translations. Which, at this stage of the game, is as good as it will get. But kudos to all the eebmg's of this world that have tamed the math beast!

On the other side, when one of the BY quizzes comes around with more general knowledge questions, I am completely out of my element.

 

[CL82]

So, if you drop a goat from a definite, but immaterial, height and no one is there to hear it, does it make a sound?

 

[and one]

The intersection of the goat and the ground at impact would create a wave pattern which if there was a detection device in the vicinity would be interpreted something like Splat. Do not attempt to try this however as goats strongly object to being dropped from even moderate heights

 

[Skeets]

You guys make me laugh, you get a "like".

 

[CocoHusky]

Did you happen to run any of this by Bria Hartley?

 

[RegisteredUconn]

So, if you drop a goat from a definite, but immaterial, height and no one is there to hear it, does it make a sound?

All depends on whether the goat is still tethered or has already eaten through the rope.

 

[RegisteredUconn]

Did you happen to run any of this by Bria Hartley?

Isn't she in the hospital undergoing "traction"? I remember that from when I was a kid and my dad had a "slipped disc." They put him in the hospital with some sort of slightly more comfortable noose around his neck and attached weights to his feet and hung them over the edge of the bed. Anyway, I don't think Bria's still the same height she used to be...whatever that was.

 

[CL82]

All depends on whether the goat is still tethered or has already eaten through the rope.

Goat hangings? That’s a little dark even for The Boneyard.

 

[MooseJaw]

The intersection of the goat and the ground at impact would create a wave pattern which if there was a detection device in the vicinity would be interpreted something like Splat. Do not attempt to try this however as goats strongly object to being dropped from even moderate heights

don't forget to take into consideration the Doppler effect.

 

[RegisteredUconn]

You speak as if all goats are homogenous and interchangeable. I'll bet there are plenty of goats that would enjoy a good dropping.

The intersection of the goat and the ground at impact would create a wave pattern which if there was a detection device in the vicinity would be interpreted something like Splat. Do not attempt to try this however as goats strongly object to being dropped from even moderate heigh

 

[RegisteredUconn]

Goat hangings? That’s a little dark even for The Boneyard.

That's your response to perhaps my lightest post ever? Anybody know of a good traction clinic for those on a thrifty budget?

 

[and one]

Isn't she in the hospital undergoing "traction"? I remember that from when I was a kid and my dad had a "slipped disc." They put him in the hospital with some sort of slightly more comfortable noose around his neck and attached weights to his feet and hung them over the edge of the bed. Anyway, I don't think Bria's still the same height she used to be...whatever that was.

I'm not the keeper of the official Hartley unit of measurement but I assumed that the unit equals Bria's height when it was declared as a unit, and any variation in Bria's height since then would not have any bearing on the official Hartley unit. In a 100 years, long after Bria as any height at all, UConn players will still be measured by the Hartley unit.

 

[eebmg]

Am I the only one who thinks goats are getting way too much attention due to a completely arbitrary and coincidental spelling of their names.

What about true warrior animals such as lions, sharks, bears etc.

 

[temery]

So, if you drop a goat from a definite, but immaterial, height and no one is there to hear it, does it make a sound?

yes

 

[and one]

OK, I'm going to step in for the mods here and warn one and all that Boneyard is no place to talk about hanging goats. Chickens perhaps but not goats.

 

[temery]

Am I the only one who thinks goats are getting way too much attention due to a completely arbitrary and coincidental spelling of their names.

What about true warrior animals such as lions, sharks, bears etc.

Ever see a goat climb a tree or the side of a mountain? Goats are incredible.

32 Photos That Prove Goats Are the World’s Best Climbers

Goats engage in increasingly bizarre and perilous feats of climbing.

matadornetwork.com