Continuity mistake: (End of Ep. 3 and start of Ep. 4) When The Third Doctor fights the dark side of Omega's mind, the rings that The Doctor wears on the pinky fingers of both hands switch to his ring fingers, then back again to his pinky fingers, between shots. Then as Omega chokes The Doctor, the ring on his right pinky vanishes between shots.
Doctor Who (1963)
1 continuity mistake in The Three Doctors - chronological order
Visible crew/equipment: (Ep. 2) When The Doctor, Jo, and Dr. Tyler are taken into Omega's palace, in the first shot of the multiple passageways (right before Jo says that it looks like Aladdin's cave) the curved chalk actor's mark is visible on the floor, then we see it again in three following shots. Note that when they're all escorted further inside, just as Dr. Tyler comments about the matter/anti-matter not making sense, Jon Pertwee looks down, presumably to make sure he hits his mark.
The Doctor: You can't rule the world in hiding. You've got to come out on the balcony sometimes and wave a tentacle, if you pardon the expression.
Planet of the Spiders - S11-E5
Trivia: The first thing the Third Doctor does on-screen is collapse out of the TARDIS, which is also the last thing he does in that incarnation.
Question: If the Doctor gets a TARDIS, does that mean other Timelords get a TARDIS as well?
Answer: TARDISes are generally available for properly authorised use on Gallifrey; they're not usually assigned to a particular Timelord on a long-term basis. The Doctor stole his when he left his homeworld.
Tailkinker ★