Plot hole: In episode four, Merak expounds on how only the Doctor and Romana can get into the TARDIS, despite the fact that he shouldn't even know what the TARDIS is. His insight continues in episode six, where he talks knowledgeably of the sixth segment of the Key to Time, despite having been told nothing about it.
Continuity mistake: When the twins start to play equations they are both facing towards us. If they were to face each other, their legs would be touching each other. During the game we again see them at this distance apart. But the third time we see them (as they finish), they have moved at least a metre apart, making it possible for them to face each other comfortably.
Revealing mistake: At various points in the series, Mr. Sin is obviously a dummy and not an actor. But the most obvious is during the last fight when Sin jumps the Doctor. After a struggle, cut to a forward angle and the Doctor grabs and throws an obvious dummy to the ground.