# The “Holey” Sphere

[From the website of Dr. Donald Simanek—see further down.]

Browsing Martin Gardner’s books I stumbled on this diabolical puzzle. Gardner calls it “an incredible problem”. He traces it back too 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.

The answer is provided by Doctor Donald Simanek, Professor of Physics Emeritus at Lockhaven University.

Visit Donald Simanek’s page at Lockhaven.edu for more brain-bending physics puzzles!

Here is The Answer from Dr. Donald Simanek
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.