For Waverly in this story the "invisible strength" is very important. This invisible strength she associates with inner strength and it is this power that helps you both respect, arguments and chess games. This is why the chess board is the important motif here and she's her and her mothers relationship as a chess game a battle for recognition and respect. Her mother because of her outburst earlier in the story has the power in their little power struggle and that's why she is pondering how to resolve her issues with her mother, and the generational gap between two of them.
Hope this helps
-Christopher