**NMSR PUZZLES**

**November 2005 Solution**

*By John Geohegan*

The November Bonus Puzzle asked for the volume remaining after
drilling a hole six inches long straight through a solid sphere. The
answer is 113.1 (36p) cubic inches, which
is the volume of a sphere having a radius of 3 inches. Surprisingly,
the answer doesn't depend on the size of the sphere other than
requiring its minimum size. A sphere larger than 3" radius R requires
a hole with radius equal to the square root of (R^{2} - 9) to
obtain the required length of 6 inches. The volume of the hole is
then 6p times (R^{2} - 9) and the
volume of the truncated spherical segment (with its hole filled) is
6p times (R^{2} - 3), which can be
found with the calculus or from a handbook which lists such formulas.
The difference (e.g. the sphere with the hole excavated) is
36p, and is independent of the sphere's
radius.