Like most AI "game" problems, there's a general approach:
Implement a tree structure where each node is the game state and children of states represent transitions between those states.
Either do this as a breadth-first search (depth-first OK if you keep a log of past states you've seen and refuse to revisit them, and you don't care about finding the optimal solution) or come up with an optimistic heuristic that allows you to use A*. A pretty-awful heuristic I can think of is "The number of circles that need to be flipped to win the puzzle, divided by 5." I'm not sure if there's a better one; I'd be interested in hearing people's input on this (Note that it has to be optimistic, that is, the heuristic can never OVER-calculate the number of moves needed.)
Going into more detail is a little silly since this is such a big topic, and besides that, it's pretty simple if you know how to do a breadth-first search or A*.
