A game of peek-a-boo with his dad excites william during the first six months of his life because william actually thinks his dad has disappeared. this is because william lacks:

William's excitement during a game of peek-a-boo in the first six months of his life is likely due to a concept known as "object permanence." Object permanence is the understanding that objects (or people) continue to exist even when they are not visible.

In the early months of life, infants typically lack a full understanding of object permanence. When an object, such as a parent's face, is covered or hidden, the infant may believe that the object no longer exists. As a result, when the parent reveals their face again during peek-a-boo, it can be surprising and delightful for the infant, as if the parent has magically reappeared.

This lack of object permanence is a normal and expected stage of cognitive development in infancy. As infants grow and develop, they gradually acquire a more sophisticated understanding of object permanence, realizing that objects and people persist even when out of sight. The development of object permanence is associated with cognitive milestones and is an important aspect of a child's early cognitive development.