While on vacation, the Beatles are warned of a jewel thief that is also a master of disguise named "Anyface". And just as soon as they are warned, there are two Pauls amongst the group. Nobody is sure which is the real Paul, so the police chief decides to have his niece, the world's biggest Beatles fan, listen to them sing so that she can pick out the real Paul.They begin singing No Reply and she decides it's the one on the left. John suggests they keep singing and then she changes her mind and says it's the one on the right. Once its decided that nobody can tell which is the real Paul, the two Pauls are asked to contest each other on the "tree of truth" near the "cave of truth". It isn't until a group of screaming fans chase them off, while Anyface remains to greet them, do they figure it out. Anyface also leaves behind his fake nose, which he was using to hide Ringo's stolen rings.
Youtube link: (skip to 3:00) http://www.youtube.com/watch?v=T_KAMsZk5i0
Remember, this episode aired in December 25th, 1965. More evidence that the entire Beatles story was pre-scripted from the beginning.
P.S. I find it very befitting that the world's biggest fan couldn't tell them apart. Sounds like every PIAer and PIDer I've ever come across.