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Cylinders in Spheres

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Re: Cylinders in Spheres

#31

Post by nemetroid »

eps wrote: post #25

There is a simpler puzzle that's similar to Gardner's, but as unintuitive. Take an orange and wrap a string around its diameter. Now extend the string by 1 inch and redistribute it around the orange so that it floats an even distance from it. Do the same with the Earth, i.e. wrap, extend by 1 inch and even out into a circle. The gap betwen the string and the orange/Earth - which one is bigger?

Spoiler below!

The relationship between radius and circumference is linear (C = 2pi * r). When the circumference is increased by 1, the radius increases by 1/2pi. Therefore, the gap has the same size.

Let g be the gap. Then

          C + 1 = 2pi * (r + g)
    2pi * r + 1 = 2pi * (r + g)
    r + 1 / 2pi = r + g
        1 / 2pi = g

Re: Cylinders in Spheres

#32

Post by maxerickson »

delluminatus wrote: post #24
dotrob wrote: post #21

Earlier quoted context omitted.

In the puzzle statement, "A six inch high cylindrical hole is drilled through the center of a sphere," through is the keyword, rather than into . I made the same initial mistake of misreading through as into.

In that case, the statement from the article "We could have a sphere as large as a planet, bore a hole 6" in length through it..." seems inconsistent. Either the cylinder is not 6" long, or it does not go through the sphere.

The problem can be rephrased to avoid the ambiguity. Something like "A hole is bored through a sphere such that the void in the remaining material has the shape of a cylinder 6 inches tall".

I guess the overall idea is anyway to reveal the elegant mathematical result. Wikipedia does a good job of talking clearly about it:

In geometry, the volume of a band of specified height around a sphere—the part that remains after a hole in the shape of a circular cylinder is drilled through the sphere—does not depend on the sphere's radius.

http://en.wikipedia.org/wiki/Napkin_ring_problem

Re: Cylinders in Spheres

#33

Post by drcode »

Someone wrote: post #27
drcode wrote: post #6

Earlier quoted context omitted.

I found it through some extra searching starting from your link- "Kopf um Kopf" ... Is that the precursor you were thinking of? Here is a video on youtube that shows the awesomeness of this show (don't need to know German to appreciate it- On the start of the show, putting your finger under the device makes it spin in the opposite direction... why? The audience member with the right answer would get a reward.) https:…

I'm not sure. Wikipedia claims that he precursor also was called Kopfball, and has some overlap in the times when the series ran. I just googled the "kopf" that I remembered in combination with "wissenschaft" (science) and "fernsehen" (television), and that popped up, and I thought "that must be it". I should have been more cautious, though. German TV at the time had quite a few interesting programs about science (in…

Yeah, Hobbythek was awesome, too... I remember at maybe 8 years old trying to rebind one of my own broken books after probably seeing that exact episode you mention...

It's really easy to get into this mode where you think "If it's not new and American it's crap" because the US in recent years has been so prodigious in creating such a large range of media of many different types, and because the US is not shy in "Americanizing" things from other countries and improving on them. However, there are definitely corners of brilliance lying in the past and in other countries that have been forgotten, and will be rediscovered in future years.

Re: Cylinders in Spheres

#34

Post by hammock »

The author never pays off the answer to the initial question- is it a fat cylinder or a skinny one? We know it has height ~1.15R (where R is sphere's radius), but this is not easy to visualize.

Re: Cylinders in Spheres

#35

Post by scotty79 »

I once encountered question that had cheat answer. It was about trapezoid and it seemed that it had some data missing.

I was amazed that just assuming that the question had one answer allowed to reduce the problem to trivially calculable one by consistently manipulating the variables that were not given.

I was also pretty proud of myself for finding this solution.

Re: Cylinders in Spheres

#36

Post by colanderman »

jerf wrote: post #8
chton wrote: post #2

The cheat is absolutely brilliant reasoning.

Ehh, sort of. It's logically flawed as phrased, in that the first bit is false: "If the problem is being posed, it must have a constant solution" is false; it could just as well be a niftily-simple symbolic solution too. (I say "as phrased" because as other commenters observe, there are mathematically valid ways to arrive at the result.) This reminds me of one of my favorite joke proofs from when I was in school. (It…

My favorite "cheat" was on an exam in an algorithm-analysis class (lots of discrete math, infinite sums, and proofs). We were asked to prove or find a counterexample to some conjecture or other. Looking at the conjecture, and looking at the clock, I decided the only thing I could complete in time was to find a counterexample. Indeed within a few minutes I found one and wrote it down.

I was the last to finish my exam and handed in my booklet; the prof looked it over and noted I was the only one (out of 5) in the class to get that particular question correct. I explained to him my "reasoning". He said, "yes, that is how you're supposed to do it!"

Re: Cylinders in Spheres

#37

Post by Chinjut »

hammock wrote: post #34

The author never pays off the answer to the initial question- is it a fat cylinder or a skinny one? We know it has height ~1.15R (where R is sphere's radius), but this is not easy to visualize.

The cylinder is sqrt(2) (~= 1.41) times as wide as it is high. So... a little fat? (I don't know the healthy baseline for cylinders...)

Re: Cylinders in Spheres

#38

Post by theophrastus »

galvan wrote: post #28
delluminatus wrote: post #24

Earlier quoted context omitted.

In that case, the statement from the article "We could have a sphere as large as a planet, bore a hole 6" in length through it..." seems inconsistent. Either the cylinder is not 6" long, or it does not go through the sphere.

I thought that at first, but the length of the hole is dependent on the width of the hole. The wider the hole, the shorter, because wider holes remove bigger caps. With a sufficiently wide hole, you could indeed drill a 6" hole through a spherical Earth, it'd just look more like a thin ring the diameter of the Earth than a sphere.

i still get a confused language impression from that. for me it's the combination of "drill" and "through" (probably was forced to take too much 'wood shop')

i think it would be more clear to phrase it starting along the lines of: position a cylinder concentric and inscribed within a sphere ...

Re: Cylinders in Spheres

#39

Post by bkcooper »

svantana wrote: post #22

Nice, I especially appreciate the "cheat answer" to the Gardner Puzzle at the bottom of the page. I have found that kind of meta-reasoning about questions quite useful, on exams and in games like Trivial Pursuit, for example.

Looking at the other comments, it seems like most people really like the cheat. I admit it's very cute, and you're absolutely right that this sort of thinking can be helpful in artificial situations like exams and games --- I've used it myself.

That artificiality is why I don't really like that approach, though. It's a brand of thinking that generally works only on artificial problems, because the key component ("you wouldn't be asking me this if it didn't have a well-defined answer") doesn't exist on most problems. Proving that it's constant and then using the r -> 0 trick to calculate the constant is much more satisfactory to me.

Re: Cylinders in Spheres

#40

Post by mdkras »

Chinjut wrote: post #30
eps wrote: post #26

Earlier quoted context omitted.

#2 is 3/Pi. It's a good puzzle and it helps to know the answer :)

Rather, 2/pi. The nice answer (referred to above as a "cheat") is to note that the sought probability is the mean number of crossings made by a 1 foot needle with the lines on the floor, where we are implicitly supposing our throw-distribution to be uniform with respect to both translation and rotation. This uniformity, along with "linearity of expectation", is such that the mean number of line-crossings from throwin…

#2 is called this: http://en.wikipedia.org/wiki/Buffon%27s_needle. This is also mentioned in Jordan Ellenberg's book How Not to Be Wrong, and has a similar passage with your explanation above. I mention the book as much because it's a great read.
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