UPDATE: Missing Pieces Located
Apparently, Big E was telling the truth when he told me they were under the rug. Shocking, I know.
I had looked under the rug in Mr. Yoy's office, the family room, and the dining room. I didn't see anything.
But Mr. Yoy had a great idea! He moved the couch and some coffee tables in the family room and we were able to see deep under the center of the rug. Sure enough, there they were! Big E must have slid the train puzzle under the rug and when he tried to pull it back out, it broke into two.
Anyway, all is right in the world again.