The Claws of Axos:

20ep 2 – The Nuton power complex contains a cyclotron which, Winser explains, can accelerate atomic particles to the speed of light and beyond. There's even an instrument which measures the effect, with a scale marked in "X light"! Unfortunately, relativity states that you can't accelerate particles to the speed of light (let alone beyond it), because at that point the particles would have infinite mass.

30ep 2 – As the Master escapes from Axos, he kills the UNIT soldier standing guard outside, whose beret falls off as he slumps to the ground. Cut to the Master standing over the corpse, and the missing beret instantly replaces itself on poor Private No-Name's lifeless cranium.

40ep 2 – Just as the Brigadier, Jo and Filer escape from the room where the army have been holding them captive, the wall of the room sways outwards (the way that a flimsy wooden wall usually does when someone leans on it).

50ep 3 – At the start of this episode, as the Axon creatures attack the nuclear lab, look out for some sloppy continuity. Just after Filer gets impaled on a tentacle and the creatures advance on our heroes, Jo is clearly standing behind the Doctor's arm. We cut briefly to a second shot (as the Axons say "repersonalise") and Jo is standing directly alongside the Doc. And finally we cut to a third shot (just before the creatures morph back into "golden girls") and Jo is plainly standing a yard in front of the Doctor.

60ep 3 – It's amazing how Sergeant Benton can keep a straight face when reporting to the Master, who is wearing the saggiest, least convincing rubber-General-disguise-mask you're ever likely to see!

70ep 4 – As the Doctor and Jo escape from Axos, Jon Pertwee trips and falls flat on his face!

90ep 4 – During the scene in the TARDIS where the Doctor offers the Master an alliance, the left-hand TARDIS door refuses to close properly. Then, when the Doctor goes to leave, despite the fact that neither he nor the Master touches the console, the right-hand door obligingly swings open for him!

100ep 4 – During the climactic gun battle in the Nuton lab, the TARDIS is briefly visible in the background, although the Doctor and Master left in the TARDIS some minutes earlier.

110ep 3,4 – In this story, the Nuton Complex is a giant nuclear power plant which supplies the whole of Britain with electricity. In episode 3, Hardiman understandably claims that if the reactor were to explode, thousands of lives would go with it, as the chain reaction would turn the station into a gigantic nuclear bomb. When exactly this event happens at the end of episode 4, not only does the reactor explode without apparent loss of life, but the Brigadier and co. calmly return to the scene of the explosion within seconds!

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