Consider the following problem:
A hole is drilled through the center of a sphere.
The cylinderwithcaps is removed.
The length of the removed cylinder (it also has caps on it which
do not count for the length) is 6 inches.
What is the volume of the remaining solid?
There are two ways to do this problem.

Here
is the solution using calculus:

Here is a solution which you may consider cheating.
The very asking of the question implies that the answer can
be determined from the data given.
Hence we can CHOOSE an instance of the problem and KNOW that our solution
for this instance is always the solution.
We choose to have a
cylinder of radius 0 (so its just is a line of length 6).
Hence the answer is the Volume of a Sphere that is
6 inches in diameter: (4/3)(π)3^{3}=36π.
There are two ways this may be considered is cheating.

Minor one: Deriving the volume of a sphere itself requires calculus
so I didn't really get around that issue.
However, the Volume of a sphere is well known so I think this is a quibble.
(Does anyone know a noncalculus proof for the formula for the the volume of a sphere?)

Major one: We used the fact that the answer can be determined from the data
to find the answer. Is this appropriate?
How would you grade this if given as an answer on an exam?
Here are some thoughts:

If you put this on an exam what would you do if a student had
this solution? Reward them for thinking outside the box or
penalize them for not showing they know calculus?

What if it was on a mathematics competition?

Best solution might be to make it a multiple choice question so they
do not need to show how they did it. Those that think of the clever solution
are rewarded by spending less time on it unless it took them a long time
to think of the clever solution. Those that do it via calculus also get it
right.
You might want to make one of the choices
Cannot be determined from the data given.

Posted By GASARCH to
Computational Complexity at 7/01/2010 07:52:00 AM