Factual error: A reflection of a full Moon is shown near the end. When Earth is viewed from the Moon earlier, over half of it is illuminated by the Sun. For it to be full Moon, the entire Earth would have to be dark. In addition, the place where the hospital was transported to was dark, and not sunlit. (00:37:45)
Continuity mistake: In the scene when Donna asks the woman if she could borrow £10 because it's Christmas, you can see that Donna is wearing sports shoes instead of her usual white wedding shoes. (00:10:05)
Suggested correction: No, she's wearing the same shoes if you pay attention.
Never mind - she is wearing running shoes for no apparent reason.
The Sound of Drums (2) - S3-E12
Continuity mistake: When the Master stands up on the Valiant to announce that he is the Toclafane's "Master", President Winters is visible in the background, only the Toclafane aren't there. In the next shot, they return, buzzing around Winters' head.
Suggested correction: This seems to be caused by angles - the Toclafane are floating fairly high up, around the level of Winters' shoulders and above, but the Master is lower down. Hence, why they're not visible in the shot.
Plot hole: The Doctor leaves Martha a recording with instructions for her for when he's human. However, the flashbacks to the incident that led to him using the Chameleon Arch don't leave enough - or any - time for him to have made the recording, presenting his decision to use the device as a spur-of-the-moment one that he immediately acts upon.
Suggested correction: He would have been planning the change to human for a while now, not just made it up on the fly. So he could have made a recording before the opening scene.
The episode makes it pretty clear that he and Martha got taken by surprise by the Family's attack. The Doctor also explicitly says, when he gets out the Chameleon Arch headset, that he never thought he'd use it. The flashbacks consistently present his decision to become human as one made as an urgent, in the moment one.
Character mistake: The question to unlock one of the doors asks for the next number in the sequence 313, 331, 367. The Doctor gives the answer as 379 with a convoluted (albeit correct) explanation of happy primes, but there are much simpler solutions. The difference between the first and second numbers is 18 and the difference between the second and third is 36 (which is 2 x 18), suggesting that the difference between the third number and the next could be either 54 (= 3 x 18), giving 421 as the next number, or 72 (= 2 x 36), giving 439. It's likely that the Doctor would see that the answer is ambiguous, and yet he confidently states that the most complicated answer is correct. (00:08:45 - 00:09:15)
Suggested correction: If you know what a happy prime is, like the Doctor does, it might be simpler than your idea because you gave 2 options - 421 and 439. The idea of a pattern is there should be only one answer that fits, which would be the case if they're all happy primes.
Suggested correction: When the reflection of the full Moon is viewed it is night time several hours after we saw the Earth from the Moon.
Peter Wallace
Yeah, no, the phases of the Moon don't work like that. New Moon and Full Moon are two weeks apart. Since the Moon appears to be, judging by the appearance of the Earth, between New and either First or Last Quarter, it is impossible for a Full Moon to be visible later that night.