Sphere puzzler

Can anyone explain the “answer” to the puzzler about taking a core out of a sphere? I don’t see how an answer is possible without knowing the diameter of the core, which is 6" long.

it said… how much of the sphere remains after you take out the 6" core?
answer - all of it. it did not say how much of the sphere is left not counting the core. just my thoughts. but what do I know.

If that’s the answer, that’s not their explanation.

the real answer explained…

  1. The holey sphere.

The holey sphere. L = 6".
From mathworld.wolfram.com.

Browsing Martin Gardner’s books I stumbled on this diabolical puzzle. Gardner calls it “an incredible problem”. He traces it back to Samuel I. Jones’ Mathematical Nuts , 1932, p. 86.It is seen on the web in various forms, often ambiguous in wording, along with endless discussions often leading nowhere. I have tried to restate it to remove ambiguity (which isn’t easy).A hole is drilled completely through a sphere, directly through, and centered on, the sphere’s center. The hole in the sphere is a cylinder of length 6 inches. What is the volume of the remainder of the sphere (not including the material drilled out)?You’d think there’s not enough information given. But there is. The solution does not require calculus. Gardner gives an insightful solution that requires only two sentences, including just one equation. Answer. Assuming the problem is fair, then there must be a solution, and since the size of the sphere and the diameter of the hole were not given, the answer must be independent of these dimensions. So consider a hole of infinitesimal diameter and length 6 inches. In that case, the remaining portion of the sphere must be the same as the entire volume of a sphere of diameter 6 inches. That is 36π, which is the correct answer, as you could verify with more mathematical drudgery.The deceptive feature of this puzzle is that one is tempted to assume that the sphere is of fixed size and remains that size as one drills different diameter holes in it. The requirement that the cylindrical hole be 6 inches in length, whatever its diameter, forces the sphere to be a different diameter for each hole size considered. One might worry that the limiting case of hole diameter zero might a problem. It isn’t. There’s no discontinuity in the function there.

found here no. 67


but in the car talk puzzler it did not say this. hence my answer above.

What is the volume of the remainder of the sphere (not including the material drilled out)?

I understand the explanation above, but that requires the length of the hole in the sphere to be 6". The Puzzler says “And the core is exactly six inches long.” To me that means the length of the piece removed is 6", not the length of the hole created in the sphere.

Sounds a little illogical to me as the outside edges of the core would be shorter due to the curvature of the sphere.

The ends of the core are flat as seen in the picture, so it would be equal to the length of the hole.
the red part remains on the sphere.

??? How do you cut out a core clean through and leave the two red parts? The puzzler solution just failed to describe what was 6" long, which was the hole, not the core.

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I use this to decore apples.

If the diameter of the hole gets bigger, the size of the sphere has to get bigger so that the top of the cylindrical core exactly matches the sphere’s surface where they meet. Bigger core/bigger sphere must cancel out. The puzzler is sort of a trick in the sense that to solve it, the solver must presume a unique solution exists w/only the given information. This concept occasionally comes up in Sudoku puzzles as well; i.e. in order to solve it, you have to presume it has been designed such that is actually possible to solve.

I expect a mathematician would say the first thing to do is show that the solution is not a function of the core size. But that part of the problem might be a little too computationally intense for the show’s general audience.