Why does Waverly see her mother as her opponent at the end of the story?
Throughout the novel, Waverly struggles to free herself from the power her mother has over her, and the chapter "The Rules of the Game" does a wonderful job explaining the origins of this. Waverly becomes a chess champion, but then eventually blows up at her mother for embarrassing her. When Waverly confronts her mother, she threatens to never play chess again, but Lindo, Waverly's mom, just turns a cold shoulder. When Waverly eventually chooses to play again, Lindo tells her it's not that easy.
The symbolism between their relationship and the game of chess is quite apparent: Waverly has the ability to beat the best of the best, but once her mother gets inside her head, she begins to doubt herself. Trying to avoid her mother's ridicule is like trying to win a mental chess match; hence, at the end of the chapter we get the "imaginary chess game" where her mother is symbolically sitting across from her, winning this game of inner strength.