The Prisoner's Dilemma

You have a puzzle toy which consists of a ring of lit buttons, each of which can glow red, yellow, or (bluish-)green, as illustrated in Figure 1. When you push a button, its color cycles in the order red $\to$ green $\to$ yellow $\to$ red. The catch is, the lights two positions away in each direction also simultaneously cycle their color in the same way. The goal is get all of the lights to glow green. Suppose they all start red. Is it possible to solve the puzzle? For a bonus challenge, describe a quick way to determine whether or not a given starting position can be solved.

_{(This problem was based on a paper by Martin Kreh, University of Hildesheim.)}