Geometrical Puzzles
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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 R2 - 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,
(R2 - 9), to twice the volume of the spherical cap, and subtract the total from the volume of the sphere, 4πR3/3. The volume of the cap is obtained by the following formula, in which A stands for its altitude and r for its radius: πA(3r2 + A2)/6.

When this computation is made, all terms obligingly cancel out except 36
π - 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 36
π.
 

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Last Update
December 7, 2003

 
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