Character mistake: The Doctor gets Jo to do some mathematics to distract her. At one point she says "49 x 2=88", when it actually equals 98, then she says that "adding 10 equals 108". (In other words, Jo (Katy Manning) fluffed her line, got one figure wrong, then went back to script...).
Plot hole: When the Doctor goes into the CET projection of Eden, to fight the Mandrells, he disappears into the bushes on the left-hand side of the screen. Cue much shaking of foliage, and Tom Baker's cries of 'ooh', 'aargh' and 'ouch'. The Doctor then reappears with his clothes all tattered and torn, due to his 'fight' with the Mandrells...but the the fact that there is not a mark (not even a scratch) on Tom Baker is something of a giveaway that all Tom Baker did was disappear behind some bushes and do a quick change into a tattered version of his costume.
Plot hole: Azmael doesn't recognise the regenerated Doctor (In the shape of Colin Baker) whereas on most other occasions other Time Lords do recognize each other through telepathy. This is odd behaviour by Azmael, considering that Azmael and the Doctor HAVE actually met before, and possibly more than once. (Azmael last met the Doctor in the Doctor's fourth incarnation, and was the best teacher the Doctor ever had. On that occasion, Azmael got drunk.).
Revealing mistake: In order to save money filming the lift-off of Crayford's rocket from Oseidon, the producers decided to use stock footage of a Saturn 5 blast-off. Unfortunately, that meant the words "UNITED STATES" in large red lettering are seen on the side of the rocket...which is a little bit of a giveaway, especially as the rocket is supposed to be a BRITISH space probe.
Other mistake: The Cyber Leader is supposed to follow the Master's footsteps exactly in order to get across the chessboard. He doesn't. This cannot be classed as a character mistake because the Cyber Leader knew what was involved. I believe this mistake was caused by the actor not being able to see the chessboard through his helmet.
Revealing mistake: Watch really closely when Leela throws the knife at one of the robots. You can see the knife fall to the floor about a second after she throws it. Then, of course, the next shot shows the knife sticking out of the robot's chest. Guess Leela isn't quite as good a shot as we thought.Matty W