Cylinders in Spheres
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Cylinders in Spheres
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Re: Cylinders in Spheres
#2The cheat is absolutely brilliant reasoning.
Re: Cylinders in Spheres
#3In the early eighties there was a TV show on German network television and I remember them presenting this puzzle, and I figured out the "cheat solution" as a kid.
It was this awesome TV game show that consisted entirely of Martin Gardner-style puzzles and other fiendish physics/biology puzzles, and contestants who were all scientists. Does anyone by any chance remember the name of this German TV show? I would really like to look up more information about it! It deserves to be remembered.
Re: Cylinders in Spheres
#4It wasn't (initially) clear to me that the cylindrical hole must enter and exit the sphere.
With that knowledge the solution seems pretty intuitive.
Re: Cylinders in Spheres
#5In the early eighties there was a TV show on German network television and I remember them presenting this puzzle, and I figured out the "cheat solution" as a kid. It was this awesome TV game show that consisted entirely of Martin Gardner-style puzzles and other fiendish physics/biology puzzles, and contestants who were all scientists. Does anyone by any chance remember the name of this German TV show? I would really…
Likely Kopfball (http://de.wikipedia.org/wiki/Kopfball_(Show)), although that is late eighties (and the precursor late seventies)
Re: Cylinders in Spheres
#6In the early eighties there was a TV show on German network television and I remember them presenting this puzzle, and I figured out the "cheat solution" as a kid. It was this awesome TV game show that consisted entirely of Martin Gardner-style puzzles and other fiendish physics/biology puzzles, and contestants who were all scientists. Does anyone by any chance remember the name of this German TV show? I would really…
Likely Kopfball ( http://de.wikipedia.org/wiki/Kopfball_(Show) ), although that is late eighties (and the precursor late seventies)
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://www.youtube.com/watch?v=1ObdE9n3UF4
Take a quick peek at a few other random spots of the show in the video and marvel at the awesome scientific experiments on live TV: I think that was one of the most bad ass shows ever, especially since it didn't have that pejorative "science shows are only for kids" thing happening.
Re: Cylinders in Spheres
#7One of those cases where using integrals rather than geometry is much simpler.
\pi \int_-3^3 (R^2 - x^2) dx = \pi (6 R^2 - 18)
\pi \int_-3^3 (R^2 - 3^2) dx = \pi 6 (R^2 - 9)
You can actually show the solution does not depend on \pi without evaluating the integral, and then compute the 'cheat' case, which would not be a cheat once you've made this observation.
Re: Cylinders in Spheres
#8The 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's so simple I'm sure it has other originations but AFAIK I independently recreated it.)
"The problem begins with the phrase 'Show that...' or similar. All previous problems that began with that phrase have been provable. Therefore, by induction, the claim I am being asked to prove must be correct. QED."
Unfortunately, this proof line hit a snag about halfway through my graph theory class in which we were assigned a problem out of the book that turned out to ask you to prove a false statement. (Well, a snag above and beyond the fact that that is an invalid inductive proof in general, ahem.) It was a typo and clearly accidental, but it was enough to break my proof forevermore. May you have better luck with it.
Re: Cylinders in Spheres
#9I remember reading and solving this the problem as the end of the article (“A six inch high cylindrical hole is drilled through the center of a sphere. How much volume is left in the sphere?”) as a kid. I did it the hard way using the formula's, but the whole point of the puzzle was what this article called the "cheat" answer. It reduces the solution to utter simplicity by application of some elegant logic. It's not something of a cheat -- it's the whole point of the puzzle.
I really was a kid. I was taken in by Gardner's April 1st column that claimed among other things that a proven solution for "Chess" indicated that white would always win and that the opening move was P-KR4 (h4 in modern notation).
Re: Cylinders in Spheres
#10It reminds me of two other neat problems -
1. Imagine a band stretched taught around the diameter of the earth (which, for the purposes of this question, is a smooth sphere). Now imagine that the band is raised one metre from the ground at every single point along its length. How much longer is it?
2. Imagine perfectly parallel lines painted on the floor, exactly one foot apart, and a rigid needle of length one foot. If you throw the needle to the floor at random, what is the probability that it crosses one of the lines? (This one has a nice 'cheat' solution just like the OP article).