The vertices of a polygon are labeled with real numbers, the sum of which is positive. At any time, you may change the sign of a negative label, but then the new value is subtracted from both neighbors' values.
Example: if three consecutive labels are 1, -2, 0, then you may change in one move the -2 to 2, the 1 to -1 and the 0 to -2.
1. No matter which labels are flipped, the process will terminate after finitely many flips, when all labels will be non-negative. The process cannot go on forever.
2. Moreover, for given initial labels, the process terminates in exactly the same number of moves regardless of choices.
3. The final configuration is independent of choices as well.