Two recent articles in this journal described how an air core solenoid connected to an ac power source may restore the magnetization of a bar magnet with an alternating magnetic field (see Figs. 1 and 2). Although we are quite accustomed to using a constant magnetic field in an air core solenoid to remagnetize a ferromagnet, it is puzzling that we can also use an alternating magnetic field to realign the magnetic domains of a ferromagnet. To make the puzzle even more intriguing, the very same setup may be used to degauss a TV screen; that is, the alternating magnetic field of an air core solenoid can also "scramble" the magnetic domains of a ferromagnet! Although the latter phenomenon intuitively makes more sense, we are still left with a confusing paradox: How can an alternating magnetic field align the magnetic domains in one ferromagnet and scramble the magnetic domains in another? Paradoxes like these are ideal for student investigations because they create a natural interest. In fact, since the two articles mentioned above were unable to explain the magnetic paradox, my students and I were intrigued enough to investigate the phenomenon and then try to understand it theoretically. This paper describes how we used the paradoxical phenomenon for a student investigation into nonlinear systems. In order to be successful, students should be familiar with calculus, preferably at the level of an advanced-placement class.