Continuity mistake: When the Doctor and Peri first land on Androzani, mountains are seen in the background. They disappear when Peri picks up the silica. A couple of shots later, they're back. (This mistake was mentioned on the DVD commentary; the mountain effect was left out for financial reasons).
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.