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.
Character mistake: After the Carrionites kill Lynley, the Master of the Revels, by using the doll to make him drown on dry land, the Doctor says, while examining the body, that he's never seen a death like it. In the Third Doctor serial "The Mind of Evil", involving a monster that could make people relive their worst fears, a character died in exactly that fashion. (00:13:10)
Factual error: The Olympic torch relay is presented as if there is only one runner carrying the torch through the entire city of London to the Olympic stadium to light the cauldron. Furthermore, when the torchbearer collapses, the TV announcer reacts as if it is now impossible to get the torch to the stadium, just before the Doctor steps in and picks it up. Any broadcast of any Olympic opening ceremonies would show that that's not how the torch relay has ever worked.
Revealing mistake: In the underground scene in the Nestene lair, the CGI used to create the Nestene is horrendous. And in a shot from behind the Doctor, the Nestene looks like a whole bunch of orange pixels.
