Posted 22 February 2008 - 08:05 AM

**sunburntpixY**,

Thanks for the inquiry to my puzzle.

I don't have a diagram that i can post. But this may be helpful...

It will be helpful to visualize the following ideas if you draw a circle on a piece of paper and think of it as a cross section of a sphere. The center of the circle is the center of the sphere. Now draw parallel lines -- equal distances from, and either side of, the center of the circle. Make sure the lines go all the way through the circle. The lines represent a cross section of a hole drilled through the sphere. The distance between the lines is the diameter of the drill. The area between the lines is a cross section of the hole. The area outside the lines but inside the circle is a cross section of the portion of the sphere that remains after the hole is drilled. And, most importantly to understand, the length of the lines between the points that they intersect the circle, from end to end, is the length of the hole. Note that the length of the hole is less than the diameter of the circle [sphere]. Repeat. The length of the hole is not the same thing as the diameter of the sphere. Finally, note that if you had drawn the lines farther apart -- go ahead and draw another pair of lines that are farther apart -- the length of the hole decreases.

OK - now think through these statements.

[1] For there to be anything left of the sphere, the diameter of the drill better be less than the diameter of the sphere.

[2] For the hole to be 6 inches long [length is measured in the direction the drill traveled while it was making the hole] the sphere had to have been at least 6 inches in diameter. You can't drill a 6-inch hole through a 4-inch-diameter sphere. But you can drill a 6-inch hole through a 7-inch-diameter sphere.

[3] In all cases, we are talking about drilling a hole completely through the sphere. You can look through from beginning of the hole to its end, and see all the way through the sphere. You could string the sphere onto a rope, so long as the rope is thinner than the drill, of course. And, in all cases, what we mean by the length of the hole is the distance between the beginning and the end of the hole. You have to see this length as being different from the diameter of the original sphere.

[3] If the diameter of the drill is negligibly small, a negligible amount of the sphere is removed, and the remaining volume is essentially the original volume: [4pi/3]r^3 and the length of the hole is essentially the diameter of the sphere. Only in this case is the length of the hole the same as the diameter of the sphere. Think of a pearl, with the tiniest of holes, string on the slenderest of threads. In the case of a vanishingly small [diameter] drill, the length of the hole approaches the diameter of the sphere, and the remaining volume approaches the initial volume.

[4] If the diameter of the drill is negligibly less than the diameter of the sphere, the remaining portion of the sphere is a tiny band of material at the "equator" of the sphere - think of the drill entering the sphere at its "North Pole" and exiting the sphere at its "South Pole", and the length of the hole - the height of the remaining volume - is now much less than the diameter of the original sphere. In the case of a vanishingly small difference between the sphere's diameter and the drill's diameter, the remaining volume approaches zero - the entire sphere has been drilled away - and the length of the hole [obviously] approaches zero as well. This would be the case if you drew your two lines tangent to your circle instead of intersecting the circle. If you really understand this point, you can see how a 6-inch hole could be drilled [somewhat disastrously to be sure] through the Earth! Remember, the length of the hole is defined as the height of the remaining portion of the sphere, not as the length of the drill. Or think of it this way: if you crawled inside the hole and painted the inside surface of the hole, the painted surface would be a circular cylinder. The length of that cylinder is the length of the hole.

Now if you can visualize all of that, you understand what "length of the hole" and "remaining volume" mean.

The puzzle is cryptically worded. Intentionally. It's meant to make the solver think through all of these things so that the question is even understood. That's part of the challenge of the puzzle.

If you can visualize the "length of the hole" then the question at least is understandable:

If you drill a 6-inch hole through a sphere, what is the volume of the remaining portion of the sphere?

And, if you've read this thread, you've heard it asserted that in every case - for every sphere whose diameter is at least 6 inches - the remaining volume is the same. A person with calculus skills can compute that volume and see that the answer can be expressed in terms of the length of the hole - only. An amazing result!

But another person - one who does not like tedious calculations - simply takes note of the fact that an answer has been requested **even though the original diameter of the sphere is not given**.

The only way that can be a reasonable request is if the answer does not depend on the original diameter. The person who reaches that conclusion smiles, rubs his/her hands together, and computes the answer for the simplest case: a 6-inch-diameter sphere. Remember, we said that's the case where the hole has zero diameter and thus removes zero volume, so that the remaining volume is the original volume:

For a 6-inch diameter sphere, V = [4pi/3]r^3 = 36pi cubic inches. And that's the correct answer.

Hope that makes this thread easier to follow. You're not alone in asking about what it means.

- bn

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell