We resolve the fate of the two original apparent horizons during the head-on merger of two nonspinning black holes. We show that, following the appearance of the outer common horizon and subsequent interpenetration of the original horizons, they continue to exist for a finite period of time before they are individually annihilated by unstable marginally outer trapped surfaces (MOTSs). The inner common horizon vanishes in a similar, though independent, way. This completes the understanding of the analog of the event horizon's "pair of pants" diagram for the apparent horizon. Our result is facilitated by a new method for locating MOTSs based on a generalized shooting method. We also discuss the role played by the MOTS stability operator in discerning which among a multitude of MOTSs should be considered as black hole boundaries.