That should be it then. You shouldn't need to do that after each move. You should do that once at the beginning of a game, and if it's solvable on the first mov

That should be it then. You shouldn't need to do that after each move. You should do that once at the beginning of a game, and if it's solvable on the first move, it's solvable for any configuration that can be reached by applying moves. Ex. if there are 4 colors, and I clicked once, I can click 3 times to get to my original state. Therefore if you prove one configuration solvable, then, given that I can always reverse my own moves to get back to that state, it means all configurations are (that can be reached in the current game). tldr: any move on a solvable grid, will result in another solvable grid.