Can you give an example of a complex math problem that is easy to solve? I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math problems (proofs, probably) that on the surface look equally challenging.  But…


*

*One is simple to solve (or prove)

*One is complex to solve (or prove) 

*And one is impossible


So if a mathematician can’t simply look at a problem and say, “I can solve that in a day, or a week, or a month, how can anyone else that is truly solving a problem? The very nature of problem solving is that we don’t know where the solutions lies and therefore we don’t know how long it will take to get there.
Any input or suggestions would be greatly appreciated.

Sometimes a problem which seems very hard turns out to be "easy" because someone was clever enough to look at it in just the correct way. Two examples of this:
a. Construct lots of non-homiltonian planar 3-valent 3-connected graphs (such graphs can be realized by convex 3-dimensional polyhedra) The Grinberg condition provides a nifty approach to finding such graphs easily: http://en.wikipedia.org/wiki/Grinberg%27s_theorem
b. Klee's art gallery problem: find the number of vertex guards that are sometimes necessary and always sufficient to "see" all of the interior of a plane simple polygon with n vertices.  V. Chvatal and Steve Fisk found simple ways to answer this question: http://en.wikipedia.org/wiki/Art_gallery_theorem
