My kitties Indigo and Beauty are especially fond of playing pat-a-paw under the door with each other - even if the door is open, they can find a way to stand on either side of the door and bat at one another from underneath it. They first learned to play this "no-see" version of pat-a-paw when they were housed in adjacent cages in the shelter from which we adopted them. That was how we knew they would get along from the start when we adopted them together, and they did. I am glad that your kitty has invented this fun game for you two to play together.