While no mathematician, I enjoyed reading about your exploration and findings, as well as the dialogs in the comments. I play Minesweeper often (set at a challenging level) to get my thinking organized. It's funny how applying an array of deductive rules to solve for unknowns (like setting up one of your series) is much more complicated than placing bombs and entering the proper numbers. I guess it's essentially the same process as solving for x in an algebraic equation -- x goes from a mystery to simply obvious.