You can actually knock that down to 8 moves (maybe less?) as the max required for a solution. You never have to press the middle square, since it flips every square, inverting the board. Inversion doesn't bring you closer to the checkered pattern of the chessboard. (An inverted chessboard... is a chessboard!)
There is a lot of math involved in this game and you can quite quickly note some less obvious rules like this one. Maybe there are some very nontrivial, but interesting things out there. I'm not good enough in combinatorics to tell... :D
8 moves are sometimes required. If you have a chessboard 3x3 grid with the middle square the wrong color, you have to flip all squares except the middle.
