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.
