
Hole in the Sphere
(solution)
If you want to avoid the calculations in Solution 1, simply take a look
directly at Solution 2 below it.
Solution 1
Let R be the radius of the sphere. As the illustration indicates,
the radius of the cylindrical hole will then be the square root of R^{2}
 9, and the altitude of the spherical caps at each end of the cylinder
will be R  3. To determine the residue after the cylinder and caps have
been removed, we add the volume of the cylinder,
6π(R^{2}
 9), to twice the volume of the spherical cap, and subtract the
total from the volume of the sphere, 4πR^{3}/3.
The volume of the cap is obtained by the following formula, in which A
stands for its altitude and r for its radius:
πA(3r^{2} + A^{2})/6.
When this computation is made, all terms obligingly cancel out except
36π
 the volume of the residue in cubic inches. In other words, the residue
is constant regardless of the hole's diameter or the size of the sphere!
Solution 2
John W. Campbell, Jr., editor of Astounding Science Fiction, was
one of several readers who solved the sphere problem quickly by reasoning
adroitly as follows: The problem would not be given unless it has a unique
solution. If it has a unique solution, the volume must be a constant which
would hold even when the hole is reduced to zero radius. Therefore the
residue must equal the volume of a sphere with a diameter of six inches,
namely 36π.
