The requirements of presidential nomination and Senate confirmation of Supreme Court nominees present two anomalies: under what circumstances can ideologically extreme nominees win confirmation and, given political polarization and the possibility of a filibuster, how are any nominees successful? This paper employs a simple unidimensional spatial model to explore these anomalies. The principal results show that little change in Court policy is possible with a single appointment, and this fact interacts with certain contexts to give the president a relatively free hand in choosing extreme nominees. Less firm conclusions are reached about the second anomaly, but the analysis sets the stage for further work on that aspect.