The Golden Ball & the Oxford Professors

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Discussion Overview

The discussion revolves around a mathematical problem involving the division of a solid gold ball between two Oxford Professors. The problem explores the volume of a cylinder that can be carved from the sphere and the implications of this division. The scope includes mathematical reasoning and calculus, with participants engaging in calculations and exploring different approaches to the problem.

Discussion Character

  • Mathematical reasoning
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that the mass of the swarf left after carving a cylinder from the sphere is 0.63 kg, but this is challenged by others.
  • Another participant calculates the volume ratio of the cylinder to the sphere as 3 / 4*SQRT(2), leading to a mass of 0.47 kg for the carved part, but expresses uncertainty about the validity of their approach.
  • Further calculations lead to a different maximum volume ratio of 1 / sqrt(3), approximating 0.58, with a corresponding mass of 0.42 kg for the swarf.
  • Participants discuss the need for calculus to solve the problem and share personal experiences with learning mathematics at different ages.
  • There is a request for detailed workings of the calculations to be shared for clarity and record-keeping.
  • Some participants share their academic backgrounds and express regrets about their career choices, leading to a meta-discussion about their professional paths.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct mass of the swarf or the validity of the various approaches to the problem. Multiple competing views and calculations are presented, indicating ongoing debate and uncertainty.

Contextual Notes

Participants express uncertainty regarding the assumptions made in their calculations and the methods used to derive their results. There are references to the need for calculus and differentiation, but no definitive resolution is reached regarding the correctness of the various mathematical claims.

Who May Find This Useful

Individuals interested in mathematical problem-solving, particularly in geometry and calculus, may find this discussion engaging. It may also appeal to those exploring the intersection of mathematics with real-world scenarios.

eldrick
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An excellent puzzle for you all - not overly hard, but a good tester :

We have a ball made of pure gold. 2 Oxford Professor had obtained it in illicit fashion during an archaeological dig in Peru. The 2 accomplices being born complexifiers, fell into a dispute about how they should divide up this valuable object.

One had a fancy to have a solid gold paperweight & as a ball is not much use for that purpose, decided that it must be a cylinder ; so he said,

" All I want is a cylinder from the ball & I can turn this up on the lathe in the laboratory. All the rest of it, the golden swarf, you shall have & you can sell it for a considerable sum. "

The 2nd Professor did some calculations & proved to his own satisfaction that any true cylinder from the sphere must contain less than half the volume of the sphere & so he agreed to the terms.

Was he wise ?

If the total weight of the of gold in the ball was 1 kg, what was the least weight of swarf his friend could make in turning a true cylinder from the golden ball ? "
 
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Is it 0.63kg?
 
topside said:
Is it 0.63kg?

nope

it's a high-school maths problem ( but probably last year of high school )
 
It's a high school maths problem for 14 year olds at most. It's a simple maths problem for people who've done maths to A-level equivalent.
 
matt grime said:
It's a high school maths problem for 14 year olds at most. It's a simple maths problem for people who've done maths to A-level equivalent.

it involves some calculus ( to get a non-calculator answer ) - they didn't teach me that as a 14y old ( only when i was 17y & doing my A -level )
 
volume(cylinder) / volume(sphere) = 3 / 4*SQRT(2) = 0.53

So the mass of the carved off part is 0.47kg

Although I solved it as a square in a circle problem and extended that answer to 3-D. I need to think more about if that's fully valid...

Happy Friday folks!


(EDIT -- I just did it for the sphere, and get the same term in the differentiation, so same answer.)
 
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berkeman said:
volume(cylinder) / volume(sphere) = 3 / 4*SQRT(2) = 0.53

So the mass of the carved off part is 0.47kg

Although I solved it as a square in a circle problem and extended that answer to 3-D. I need to think more about if that's fully valid...

Happy Friday folks!


(EDIT -- I just did it for the sphere, and get the same term in the differentiation, so same answer.)

Something's wrong. I get [tex]max(\frac{V_{cyl}}{V_{sphere}}) = \frac{1}{\sqrt{3}}[/tex] which approximates 0.58. Second part is 0.42.
 
Curious3141 said:
Something's wrong. I get [tex]max(\frac{V_{cyl}}{V_{sphere}}) = \frac{1}{\sqrt{3}}[/tex] which approximates 0.58. Second part is 0.42.

correct !

can you give us the detailed workings - it's nice for the public record :smile:
 
eldrick said:
it involves some calculus ( to get a non-calculator answer ) - they didn't teach me that as a 14y old ( only when i was 17y & doing my A -level )

Really ? I self-learned Calc (up to basic Integral Calc, including Volumes of Revolution) "for fun" when I was 11 or so. School taught (bored) me with it again at 15-16 before O levels.
 
  • #10
eldrick said:
correct !

can you give us the detailed workings - it's nice for the public record :smile:
Yeah, what'd I do wrong, I wonder. Did you get

0 = d/dTheta ( sin(theta)cos(theta) ) ?
 
  • #11
eldrick said:
correct !

can you give us the detailed workings - it's nice for the public record :smile:

Taking a section thru the middle of the sphere/cylinder it's obvious that to have any hope of maximising the cylinder's volume, you'll need to inscribe it symmetrically into the sphere (touching the inside).

Then, in the cross-section, it just becomes a simple rectangle within a circle problem. Let the height of the cyl. be h Then the radius r of the cylinder is [tex]r = \sqrt{R^2 - \frac{h^2}{4}}[/tex]

[tex]V_{cyl} = \pi r^2h = \pi(R^2 - \frac{h^2}{4})(h)[/tex]

Differentiate that wrt h, set it to zero and solve for h, then find V_cyl, divide by 4/3*pi*R^3.
 
  • #12
berkeman said:
Yeah, what'd I do wrong, I wonder. Did you get

0 = d/dTheta ( sin(theta)cos(theta) ) ?

I did it trigonometrically as well, if you're using the same theta I'm using, there should be a square on the sine term.

More precisely, if [tex]\theta[/tex] is the angle subtended between a vertical line drawn from the center of the cylinder and a line drawn from the center of the cylinder to the point where the limiting disk of the cylinder touches the circumscribing sphere, then

[tex]V_{cyl} = \pi R^2\sin^2\theta (2R\cos\theta) = 2\pi R^3\sin\theta \sin{2\theta}[/tex]

max V_cyl when [tex]\tan{\theta} = \sqrt{2}[/tex]
 
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  • #13
Curious3141 said:
Really ? I self-learned Calc (up to basic Integral Calc, including Volumes of Revolution) "for fun" when I was 11 or so. School taught (bored) me with it again at 15-16 before O levels.

unfortunately for me, my library didn't stock maths primers then, i had to wait 'til i was 14y ole, won the local rotary club essay prize - tasted vol-a-vents for the 1st time, got a £15 book token - with which i bought my 1st maths book - boz & chaz

more unfortunately, i didn't get a chance to do a maths degree after - i ended up as a cardiologist :frown:
 
  • #14
Could someone post the detailed workings from a - z , in "code" form - nice for the record ! :approve:
 
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  • #15
eldrick said:
unfortunately for me, my library didn't stock maths primers then, i had to wait 'til i was 14y ole, won the local rotary club essay prize - tasted vol-a-vents for the 1st time, got a £15 book token - with which i bought my 1st maths book - boz & chaz

more unfortunately, i didn't get a chance to do a maths degree after - i ended up as a cardiologist :frown:

You're a cardiologist ? I'm a Clinical Microbiologist ! Well, in training anyway. I've always felt I've missed my true calling by becoming a Physician.
 
  • #16
Curious3141 said:
You're a cardiologist ? I'm a Clinical Microbiologist ! Well, in training anyway. I've always felt I've missed my true calling by becoming a Physician.

I assume you mean you missed your true calling by not becoming a Physician , as a Microbiologist isn't considered as having missed their true calling if they pass their MBBS or MD & then specialise in Microbiology

A Clinical Microbiologist without a prior MBBS or MD , is not a Physician
 
  • #17
eldrick said:
I assume you mean you missed your true calling by not becoming a Physician , as a Microbiologist isn't considered as having missed their true calling if they pass their MBBS or MD & then specialise in Microbiology

A Clinical Microbiologist without a prior MBBS or MD , is not a Physician

I have an MBBS. I am a Clinical Pathologist.:rolleyes:

My greatest regret is not going to CalTech (I had admission and a scholarship). My parents wanted me to do Medicine locally.

I consider my true calling to be within the Physical Sciences, pure or applied. I regret doing Medicine, which I consider to be intellectually unstimulating and a waste of time. Pursuing a non-clinical, academically-oriented discipline as a postgraduate is a compromise, making the best of a bad deal.
 
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  • #18
Curious3141 said:
I have an MBBS. I am a Clinical Pathologist.:rolleyes:

My greatest regret is not going to CalTech (I had admission and a scholarship). My parents wanted me to do Medicine locally.

I consider my true calling to be within the Physical Sciences, pure or applied. I regret doing Medicine, which I consider to be intellectually unstimulating and a waste of time. Pursuing a non-clinical, academically-oriented discipline as a postgraduate is a compromise, making the best of a bad deal.

More intriguing !?

MBBS is a "English" degree

They don't offer it to the Far East Guyz unless they study in England for at least 3y post-clinical...
 
  • #19
eldrick said:
More intriguing !?

MBBS is a "English" degree

They don't offer it to the Far East Guyz unless they study in England for at least 3y post-clinical...

Please educate yourself on what Medical Degrees most former British colonies offer in their Universities. Singapore is a former Brit colony, FYI.
 
  • #20
Ah, you wanted the exact amount cut off, rather than verifying that the professor in question was correct. I think there is a non-calculus method for doing it, but can't remember it. It's a famous problem, and certainly appears in the books of either Martin Gardner or David Wells.
 

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