6 ms·

It wasn't (initially) clear to me that the cylindrical hole must enter and exit the sphere. With that knowledge the solution seems pretty intuitive.

thedufer 12y ago
Nor that the endcap(s) would not be included in either the 6 inches or the resulting volume. It seems to me that there must be a much better phrasing that brings these points home, although it might require a diagram.
TheLoneWolfling 12y ago
I've always heard it described as a circular napkin ring where the depth of the inner hole is an inch.
fleitz 12y ago
How can one drill a 6" long hole through a sphere of more than 6 inches in diameter? What I mean is if you drill 6 inches into the earth, you haven't passed through the other side... edit: I think they mean drill a 6 inch hole of maximum width, which of course would just leave a very thin ring of the earth 6 inches tall.
- dotrob 12y ago
  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.
  - delluminatus 12y ago
    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.
    - galvan 12y ago
      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.
      - theophrastus 12y ago
        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 ...
    - maxerickson 12y ago
      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 http://en.wikipedia.org/wiki/Napkin_ring_problem
- millstone 12y ago
  It doesn't say "through the sphere." It says "through the center of the sphere." For example, drill a thin radial hole 6" inches deep through the center of a sphere of radius 5"; you will pass through the center and stop before you reach the far side. The volume remaining then depends on the radius of the drill bit, and the size of the sphere. It also doesn't even state that the hole must enter the sphere. A large solid sphere that internally contains a six inch hole through its center would qualify too. This was my objection when I first encountered the problem. Everyone else seemed to understand that the hole must pass through both sides of the sphere, but that's not stated or even implied in the problem.

thedufer 12y ago

Nor that the endcap(s) would not be included in either the 6 inches or the resulting volume. It seems to me that there must be a much better phrasing that brings these points home, although it might require a diagram.

TheLoneWolfling 12y ago

I've always heard it described as a circular napkin ring where the depth of the inner hole is an inch.

fleitz 12y ago

How can one drill a 6" long hole through a sphere of more than 6 inches in diameter? What I mean is if you drill 6 inches into the earth, you haven't passed through the other side... edit: I think they mean drill a 6 inch hole of maximum width, which of course would just leave a very thin ring of the earth 6 inches tall.

dotrob 12y ago
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.
- delluminatus 12y ago
  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.
  - galvan 12y ago
    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.
    - theophrastus 12y ago
      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 ...
  - maxerickson 12y ago
    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 http://en.wikipedia.org/wiki/Napkin_ring_problem
millstone 12y ago
It doesn't say "through the sphere." It says "through the center of the sphere." For example, drill a thin radial hole 6" inches deep through the center of a sphere of radius 5"; you will pass through the center and stop before you reach the far side. The volume remaining then depends on the radius of the drill bit, and the size of the sphere. It also doesn't even state that the hole must enter the sphere. A large solid sphere that internally contains a six inch hole through its center would qualify too. This was my objection when I first encountered the problem. Everyone else seemed to understand that the hole must pass through both sides of the sphere, but that's not stated or even implied in the problem.