Four to Doomsday:

10ep 1 – Two minutes into episode 1, look out for a cameo from a boom mic, appearing above Nyssa's head in the top-left corner of screen.

20ep 1 – As the Doctor leads the others out of the TARDIS, the doors open for them without anyone going near the console.

30ep 1 – In the room where the TARDIS lands, Tegan says "Look!" and points to the top of the stairs — but only then does the door she's supposed to be reacting to actually open!

40ep 1 – Tegan's sketch of "Earth fashions" is awfully quick — and awfully good. If she's that talented, what's she doing as an air stewardess?

50ep 1 – When Monarch tries to open the TARDIS, see if you can spot the head of a crewmember hiding behind the console in the foreground, containing the lasers commanded by Monarch.

60ep 1 – How come Bigon and Lin Futu both speak excellent English, but the Aborigine doesn't?

70ep 1 – Out of literally thousands of Aboriginal languages that have ever existed, it's astounding that Tegan can speak the correct one — especially since the language in question is 12,000 years old!

80ep 2 – The Doctor says, "...if a frog with a funny head can turn itself into a semblance of a human being in a matter of minutes, there isn't much of a limit to what it can't do". The last phrase actually means the opposite of what the writer presumably intended (that there isn't much of a limit to what it can do).
[Terrance Dicks corrected this error in the novelisation]

90ep 2 – The Doctor says "the Mayans in South American flourished eight thousand years ago". Oops — actually, their civilisation began around 2,000 BC and (historians generally agree) reached its peak in the early centuries AD. So when they "flourished" was more like 2 thousand years ago... 3 or 4 at a push. Not 8!
[Bit sloppy of the Doc to make this mistake, considering he's probably been to visit them at some stage!]

100ep 2 – One of the Greeks watching the Mayan dance from the balcony is wearing distinctly un-Ancient-Greek white trainers (i.e. sneakers).

110ep 3 – When Nyssa is led out of the throne room to the Mobiliary by two of the Greek warriors, the reflection of a crew member is briefly visible as the door opens, in a panel of the corridor outside.

120ep 3 – When the Greek warriors escorting Nyssa leave the throne room, they are not wearing their helmets. In a shot of them leading her down a corridor, however, the helmets have mysteriously appeared. When they eventually arrive in the Mobiliary, the helmets are gone again.

130ep 3 – On two separate occasions, the Doctor uses his sonic screwdriver to send a monopticon spinning around rapidly, both times in an anti-clockwise direction. However, the monitors in Monarch's control room show the images as if the monopticons were spinning in the opposite direction (clockwise).

140ep 4 – When the Doctor is convincing Lin Futu to help him, watch the extra in the background as Lin Futu is asking, "How can I re-circuit Bigon without Monarch knowing?" He's just lounging there, casually watching the actors until he realises the camera's on him, whereupon he quickly starts working away again at his console!

150ep 4 – Look closely at the Chinese Dragons to spot the dancers underneath, dressed in their most authentic Ming Dynasty jeans and T-shirts! Even more amusingly, a Kung Fu Club sign is plainly visible in one close-up...

160ep 4 – After the Doctor jumps out of Monarch's ship, he drifts towards the TARDIS, slows down, and stops. Since there's no air resistance (or other sources of friction) in space, the Doctor should have continued moving towards the TARDIS at a constant speed.

170ep 4 – And continuing the theme of dodgy physics, the scene with the Doctor space-walking and using a rebounded cricket ball to propel himself back to the TARDIS must have Isaac Newton spinning in his grave! For one thing, the Doctor couldn't have been propelled backwards that fast by such a relatively small object as a cricket ball. What should happen, if the Doctor were to actually try it, would be this: after releasing the ball in the first place, he would have started moving backwards by the action of throwing the ball (remember Newton's Third Law?). And not just moving in a straight line, mind you — he would have spun backwards in a slow cartwheel as a result of pitching the ball cricket-style, as he did. Assuming he was lucky enough to get a perfectly perpendicular bounce from that spacecraft, the ball would have caught up with him, impacting with whatever part of his body was facing that way at the time, increasing the rate of his spinning motion (but not by much, certainly nowhere near as fast as portrayed on-screen).

180ep 4 – When the Doctor and Tegan exit from the TARDIS in the recreational chamber, they run off and leave the TARDIS door wide open — right in front of Monarch! Not only that, but when they return to the recreational chamber for the final confrontation with Monarch, the TARDIS doors have magically closed.

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