Daemon in a Pentagon

There is a problem in the first chapter of the course notes called Daemon in a Pentagon. It has nothing to do with some mythical creature living in a Washington military complex, rather it is like one of those puzzles in the paper, that grabs hold of you and won't let go.

I will quote the original text: (Ch1, p10.)

There is a pentagon, and at each vertex there is an integer number. The numbers can be negative, but their sum is positive. A daemon living inside the pentagon manipulates the numbers with the following atomic action. If it spots a negative number at one the vertices, it adds that number to its two neighbours and negates the number at the original vertex. Prove that no matter what numbers we start with, eventually the daemon cannot change any of the numbers.

I am going to try to solve this problem, following the G.Polya, how to solve it guide.

http://www.math.edu/~alfeld/math/polya.html

First you have to understand the problem . . .

Hey, if I understood the problem, I could solve it right away? but it is true that if you don't know the problem, if you aren't familiar with the problem, you will probably dither away your time, rather than solve the problem.

I tried doing several "daemon pentagons" on paper to see what would happen. The worst possible example is included here: