Utopia / The Sound of Drums / Last of the Time Lords:

10ep 1 – In the pre-credits sequence, the Doctor refers to his previous adventure in Cardiff as "trouble with the Slideen" — er, don't you mean "Slitheen" Doc?

20ep 1 – If Jack dives headlong (i.e. horizontally) at the TARDIS, how exactly does he end up fully upright while "hugging" it in the vortex? Shouldn't he have banged his head crashing into it?

30ep 1 – So our heroes have travelled 100 trillion years into the future (that's 1 followed by 14 zeroes) — while it's easy to say, that's an almost literally unimaginable amount of time. To put it in perspective, the universe is currently 13.7 billion years old, which sounds a lot (and it is!) but 100 trillion is over 7,000 times older than that! (No wonder the Time Lords didn't venture that far!) Why am I bleating on about how far in the future it is? Well, just to underline how unlikely it is that so much technology from 20th/21st century Earth has seemingly survived unchanged that long. For example, the AK47-style machine guns used at the start of the episode, QWERTY keyboards (visible when Jack enters the radiation room), the UK-style fuseboxes (visible when the Futurekind "stowaway" vandalises the power supply), etc. etc.!

40ep 1 – During the first scene with Yana and Chantho: "I am happy drinking my own internal milk." "Yes, well... that's quite enough information, thank you." You could buy this conversation if they'd just met, but they've been working together for seventeen years! Surely he'd know about this already?
[OK, you could argue that he already knew about it, but didn't like being reminded of it. As if she were discussing her menstrual cycle perhaps?]

50ep 1 –
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Glad to see that Health & Safety regulations still apply in the year 100 trillion...
During the "Show me your teeth" scene, keep your eyes on the right-hand side of the screen as the gate is opened to see a quick flash of a crew member in a hi-visibility jacket standing behind the fence!
[Apparently this is only visible when you're watching in full 16:9 widescreen, not 14:9]

60ep 1 – Why don't the guards bother to re-lock the gate after scaring the Futurekind away? Instead they just casually stand there with the gate ajar, showing no interest in securing the only barrier between them and the hostile creatures outside.

70ep 1 – After the "standoff" at the gate, as the Doctor & co are led inside, the front of Tennant's suit is clearly splattered with mud, yet when he arrives inside — what a surprise — it's clean again!

80ep 1 – Professor Yana is surprised that the Doctor doesn't know where Utopia is because "every human knows of Utopia". But only a minute previously he established that the Doctor wasn't human! Short memory, Prof?

90ep 1 – Professor Yana admits that his title is an affectation because "there hasn't been such a thing as a university for over a thousand years!" Now, stop and think about that: they're in the year 100 trillion, so how likely is it that universities lasted 99.999999999 trillion years (that's 100 trillion minus 1 thousand) and then suddenly gave up the blink of an eye (relatively speaking) before this episode's setting?
[Answer: Not bleedin' likely!]

100ep 1 – First the Doctor realises that Yana has to stay behind to activate the engine from outside the rocket, so he can't travel to Utopia. Then a moment later, when the Doctor learns that the TARDIS has been found, he proclaims "It's a wild stab in the dark, but I might just have found you a way out", i.e. a way to let Yana go on the rocket after all. And then we see him doing some sort of technical jiggery-pokery involving running cables out of the TARDIS control room. But why does he bother with all that? Now that the TARDIS is back, why doesn't the Doctor just let Yana press whatever button he needs to, then simply whisk him off to Utopia (or to the rocket) using the TARDIS? But no, he carries on with whatever overcomplicated scheme he has in mind. And then after all that, he eventually kicks off the launch process without even checking if Yana is on board the rocket!
[Yes yes, I realise RTD needed the cables running out of the TARDIS as an excuse to leave the door open, so that Yana could easily get into the TARDIS after his "transformation". Would be nice if he could have found a logical excuse though!]

110ep 1 – The design of the radiation room is astonishingly unlikely — just a single sliding door to get in or out? In reality you'd have a two-stage / airlock-type door to avoid giving the outside world a nice blast of radiation every time someone enters and leaves the room!
[Although since this is "magic" radiation that only attacks flesh and leaves everything else intact, maybe it's also polite enough to wait inside the door unless invited out!]

120ep 1 – When Jack manages to "fix" the couplings enabling the rocket to be launched — it's all systems go as the Doctor sets the countdown for 2 minutes and then runs around like a headless chicken to make sure all is ready for launch. But all that urgency: what's it for? Why, after years of waiting, is it vital the humans take off this very instant?

130ep 1 – Also notice that after Jack emerges from the fuel chamber, he's wearing his white T-shirt as he starts helping the Doctor with the imminent launch. But less than 15 seconds (screen time) later, he's somehow found time during the hectic launch preparations to put his navy blue shirt back on, all neatly buttoned up and tucked into his trousers!
[Never let it be said that Jack Harkness failed to dress for an occasion...]

140ep 1 – When he's opening the watch, Yana clearly has his back to the TARDIS. But in the next shot we see of him, he has inexplicably rotated 180 degrees and is facing it instead.

150ep 1 – When the Futurekind storm the (clearly chained and padlocked) gates, marvel at how easily they swing open!

160ep 1 – It's never stated definitively whether the Futurekind are human: just Yana's speculation that they might have evolved from humans. But whether they are or not, clearly they're intelligent — although their manners & dentistry leave a lot to be desired — so why doesn't the Doctor show any compassion whatsoever for them? No, instead he just runs away from them at the start & ultimately dooms them to extinction by leaving them behind when the rocket blasts off at the end. He doesn't make a single solitary effort during the episode to communicate or reason with the Futurekind — that's not how the Doctor normally treats intelligent creatures!

170ep 2 – Right at the beginning of The Sound of Drums (after the Doctor, Jack and Martha have teleported), Martha is leaning against the wall with one hand on her forehead and the other on her stomach. As we watch, she straightens herself out and lowers her hands as she moves away from the wall, but the next camera shot has her instantly back holding her head and stomach again.

180ep 2 – The Doctor's stunned realisation that "The Master is the Prime Minister of Great Britain" is undermined slightly by him getting his terminology wrong: it should be "Prime Minister of the United Kingdom" since the UK also includes Northern Ireland (which the term "Great Britain" does not).
[The Doctor is not only an alien but also in shock, so perhaps we can let him off the hook for that little slip. However... RTD, who wrote the script, and Tennant, who read the line, have no such excuse!]

190ep 2 – Watch closely when the Master throws his official Cabinet papers into the air: in long shot, they clearly fly up & to John Simm's left — but in the close-up, they flutter down all around him.

200ep 2 –
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Sorry, I can't think of anything amusing to say about a magically appearing piece of paper...
Also, during the rest of the Cabinet scene, keep an eye on the black speakerphone on the table in front of Saxon: depending on the shot, it alternates from being fully visible to being covered by a sheet of A4 paper across its top.

210ep 2 –
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I am the Master... of turning gas masks upside down without touching them!
When the Master puts on his gas mask, the round metal "V" logo is above the air tube, and upside down. In the next shot, the logo is below the tube, and the right way up.

220ep 2 – Just after the gassing scene, as we pan from the dead minister's body to Saxon drumming on the tabletop with his right hand, you can see through the glass table to John Simm's left hand, which he seems unsure what to do with — first it's resting on his left thigh, then he quickly starts moving it towards his other leg, but then he thinks better of it & rests it back on his left thigh again.

230ep 2 – So Saxon makes one cryptic reference to "medical student" in his speech and the Doctor within a split second deduces that there's dynamite behind Martha's television set? Huh??

240ep 2 –
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Be vewwy vewwy quiet, I'm hunting Time Lords...
Skirting over the fact that the dynamite prop looks too cartoonish (like something out of Looney Tunes), I'll just draw your attention to the fact that the yellow wire closest to the camera is just sticking out & seemingly isn't connected to anything!
[Also — dynamite? Who uses dynamite these days? This is the 21st century: explosives technology has moved on from the days of Wile E. Coyote! A slab of C-4 would've been more likely]

250ep 2 – The exterior of Martha's flat doesn't match the interior. When the dynamite goes off and we see the bay window explode, notice that the house's front door is to the left of the bay window, hence the stairs and landing would be too — in other words, from the perspective of someone in the upstairs flat looking out the window, the entrance door should be on the right-hand side of the room (or possibly at the back of the room, depending on the upstairs layout). However, the interior of Martha's flat we saw earlier clearly had the entrance door on the left side of the room.

260ep 2 –
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(Insert cliché about men never stopping to ask for directions)
When Martha, the Doctor and Jack are racing to save Martha's family, they pass a blue van on their left, parked outside a house with very distinctive brown & grey brickwork. It's noteworthy because only a minute later when supposedly driving away from the Jones house, they pass the same van and house, with both of them still on their left! Are they driving in circles?

270ep 2 – In the same scene (during the rush to the Jones house), our heroes pass a dustcart/council lorry which clearly has a Welsh Red Dragon on the front. What's a Welsh council vehicle doing in "London"?
[Maybe with all the alien invasions decimating London, they needed to borrow some bin lorries in a hurry and Wales had some to spare?]

280ep 2 – It's impressive that Martha's seemingly ordinary car can escape from being fired on by multiple assault rifles, with only its rear window getting broken and no damage whatsoever to the rest of the car — the bullets simply "bounce" off the side, not even causing a scratch!
[Forget tanks — looks like the Army should invest in a fleet of Vauxhall Corsas to drive into battle — those things must be impregnable!]

290ep 2 –
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Is this an elaborate double bluff to throw Saxon off the scent?
Martha's brother says he's in Brighton, but anyone who's ever been to Brighton will instantly realise he's telling porkie pies as Brighton doesn't have cliffs along its seafront.
[This scene was actually filmed in Penarth, a seaside resort close to Cardiff]

300ep 2 – The American news anchor announces that "it's 3 a.m. in the morning on the Eastern seaboard". As opposed to 3 a.m. in the afternoon?

310ep 2 – Winters introduces himself to the Toclafane as "Arthur Coleman Winters, President-elect of the United States of America" — unfortunately "President-elect" has a very specific meaning: a Presidential candidate who has just won the election, but not been sworn in as President yet. In other words, a President-elect technically isn't President of the US and so has no actual authority! It's pretty clear from the rest of the episode that Winters is meant to be the "real" US President: people address him as "Mr. President", he flies in Air Force One and talks about using his seal of office, none of which would apply to a President-elect!
[So it would appear RTD goofed up on the script side — presumably he thought "President-elect" sounded suitably grandiose without stopping to look up what it meant?]

320ep 2 – So Saxon orders President Winters killed, which he duly is — but what are the US Secret Service doing during all this? Yes, it happens very quickly, but they seemingly make no effort whatsoever to shoot/subdue Saxon or the Toclafane and instead just stand around meekly while Saxon's men hold them at gunpoint, without retaliating in the slightest.
[Anyone who's seen the video footage of the Ronald Reagan assassination attempt will know how quickly and forcefully the Secret Service spring into action when their man is threatened or shot at!]

330ep 2 – David Tennant's diction lets him down: on more than one occasion, the way he says "Master" confusingly sounds just like "Martha"! For example, after Winters is killed and the Doc pleads "Master... just calm down. Just look at what you're doing!"

340ep 2 – So it's made perfectly clear that the arrival of the Toclafane is happening at 8am sharp and the activation of the paradox machine is timed for exactly 2 minutes past 8. The only problem is that the whole scene after the Toclafane appear takes twice as long as this (pretty much exactly 4 minutes), until the Master checks his watch and mutters "Two minutes past".

350ep 2 – When the Master proclaims "Here come the drums!" and runs to the porthole to watch the rift opening, he pushes against the wall, which visibly wobbles.

360ep 3 – At the very beginning of Last of the Time Lords, the computer announces that "Sol 3 is now entering terminal extinction" Nasty. That would be as opposed to the temporary sort of extinction, then?
[Note: this is only visible in the full 50-minute version of the episode as originally broadcast on the BBC. Depending on how you're viewing the episode, you may be watching the slightly edited 45-minute version, which does not contain this scene]

370ep 3 – Jack's been chained up for 365 days — and as you can see, his face and arms are absolutely filthy from lack of care — yet somehow he's completely clean-shaven, without a single hint of stubble! And his hair, while unbrushed, isn't long and shaggy like someone continuously imprisoned for a year would be.
[You could argue "well, maybe his hair not growing is a side effect of immortality?" But Jack himself mentions at the end of the episode that he's been getting "the odd little grey hair" — so his hair must still grow!]

380ep 3 – Why are Martha's family (and Jack) magically immune to the effects of the Archangel satellites?

390ep 3 – As Jack is preparing to break out of his chains, the LCD clock on the Valiant bridge shows the time as "14:59" but the date in the lower left corner says "1 1 SA", i.e. Saturday January 1st. Looks like the production team didn't bother to set the date, just the time!
[This blooper is not visible in the 45-minute edit of the episode]

400ep 3 – When The Master "ages up" the Doctor, look in the background to see Mrs. Saxon's arms and hands "flicker" as she stands there watching, sadly betraying the fact that the aging effect was achieved by speeding up the video footage.

410ep 3 – When Professor Docherty is analysing the data from the CD-ROM, she reads that the lightning strike "transferred charge of 510 megajoules". Unfortunately, electrical charge (such as that produced in a lightning strike) is measured in coloumbs, rather than joules. OK, yes, you could convert it, since joules can be used to measure any kind of energy (and electricity is a form of energy, of course) but then she should have said "transferred energy of 520 megajoules", not "charge".

420ep 3 – After Milligan fires his gun to get the Toclafane's attention and then runs for cover, his gun switches instantly between camera angles: from his right hand to his left hand... and then back to his right hand again.

430ep 3 – Watch out for the amazing disappearing "boardroom table" on the Valiant: In the early part of the episode it's clearly visible taking up much of the lower level of the Valiant's bridge. But then when Martha is captured and brought aboard, it has disappeared!

440ep 3 – When the Master ages the Doctor into his "shrunken" state, his suit is clearly shown as not being affected by the aging rays (as you'd expect, it only works on living tissue). So for the rest of the episode they dress him in a miniature version of the suit (to avoid having to show a naked Gollum-type creature before the watershed I suppose!) But at the end when the Doctor "grows" back to his normal self, somehow the mini-suit he's in manages to stretch along with his body! Not to mention the fact that the cage he's in magically disappears at the same time!
[Sorry to disappoint the Tennant fans hoping he'd do an Incredible Hulk act and end up with torn clothes — but clearly there were a few earth persons who were very fashion-conscious and chanted "The Doctor's clothes... the Doctor's clothes..." hence restoring them too?]

450ep 2,3 – So the big reveal at the end of the episode is that the Doctor whispered in Martha's ear to "use the countdown" so humanity would know when to think of the Doctor. But at the time he said this (back in The Sound of Drums) how could the Doctor have possibly known there was going to be a countdown? He didn't know what the Master's plan was back then!

460ep 3 – Milligan states early in the episode that Martha was "the only person to get out of Japan alive" and Tish reinforces this later by recounting that the Master made them stand on the Valiant and "watch Japan burning", but at the end where we see people from across the world chanting the Doctor's name, one of the images shown is of the famous and very distinctive Shibuya zebra crossing in Japan (the world's busiest pedestrian crossing, apparently) with the residents of Tokyo none the worse for wear despite their country seemingly burned to the ground!

470ep 3 – Martha's trek around the world to "spread the word" about the Doctor, while a suitably epic conclusion, fails all tests of basic credibility if you stop and think about it even for a moment.

Mainly, how could one person coordinate billions of people across the planet to think the same thing at the same moment? Especially when she's reliant on word-of-mouth instead of mass media? First and foremost — she obviously didn't talk to 6 billion people herself (she'd need a lot longer than a year for that!) but relied on others to pass along the message. Fine in theory, but have you ever played the game "Chinese Whispers"? (Americans will know this as the game of "telephone".) If you have, you'll realise that trying to pass along a message intact from person to person to person is surprisingly difficult with 10 people in a single room who all speak the same language... never mind doing the same with 6 billion people spread across the world who speak hundreds or thousands of different languages!

And even if she got the message across accurately, who's to say they'd believe her? If you met a woman who told you that 6 or 12 months from now, when you hear a countdown, to think of the word "Doctor" and humanity will be saved: most people would a) dismiss this as the rantings of a madwoman and/or b) not even remember to do it! (A year is an awfully long time, especially when you're in a desperate struggle for survival every day in a world overrun with Toclafane)

Martha and the Doctor would have been lucky to have six thousand people chanting at the correct time, much less six billion!
[Maybe the Doctor used his psychic connection to the Archangel network to help remind humanity of the right moment to think of his name? Still though...]

480ep 3 – The special effects crew put a lot of work into the effect of the Doctor floating over the stairs towards the Master after being freed from his cage. A little too much work in fact! As he passes over the top of the stairs, he casts both a shadow and a glow on the floor — stop and ask yourself how this could be possible...!

490ep 3 – When Jack gets into the TARDIS and opens fire on the paradox machine, there are a couple of awkward moments where we hear the sound of machine gun fire, but no flashes appear from the muzzle of his gun.

500ep 3 – When the paradox machine is destroyed and time runs backwards, the view from space shows the Earth rotating east to west, which would be correct, but the Moon is still orbiting west to east, which would not.

510ep 3 – After time is reset to the previous year, the Doctor confirms that they're back to 8:02am: the moment the paradox machine was switched on in the first place (i.e. just after President Winters was killed in The Sound of Drums). To confirm this, the Doctor turns on the radio and we hear UNIT asking "what's happened up there? We just saw the President assassinated!" So far, so good, but stop and think for a sec — if time really has been rewound to just after Winters was killed, what happened to all the others who were on the bridge of the Valiant at 8:02am in the previous episode? (journalists, security, etc.) Not to mention the four Toclafane who actually killed Winters in the first place!
[Even if you argue that the paradox machine prevented the Toclafane travelling back in time in the first place, then shouldn't Winters be alive as there was nobody to kill him...?!]

520ep 3 – When Martha calls Dr Milligan, he answers the phone dressed in a white coat — but earlier in the episode Milligan stated that he was "in paediatrics". So what? Well, I've been assured by Those In The Know that paediatricians in the UK never, ever wear white coats (in fact generally don't wear a uniform at all) — this apparently is to make it easier to build a rapport with their young patients, i.e. lack of a uniform makes them seem more approachable (or something).
[This scene is not present in the edited 45-minute version]

530ep 3 – At the end, as the camera pans up from the Titanic lifebelt on the floor to the Doctor's face, just for an instant you can see Tennant looking straight into the camera (presumably waiting for a cue?) before he instantly realises the camera is on him and he looks down at the lifebelt and carries on with the scene.

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