Many remarkable properties related to chaos have been found in the dynamics of nonlinear physical systems. These properties are often seen in detailed computer studies, but it is almost always impossible to establish these properties rigorously for specific physical systems. This article presents some strange properties about basins of attraction. In particular, a basin of attraction is a "Wada basin" if every point on the common boundary of that basin and another basin is also on the boundary of a third basin. The occurrence of this strange property can be established precisely because of the concept of a basin cell.