The Lazarus Experiment - S3-E6
Continuity mistake: After Lazarus switches his machine on while the Doctor and Martha are inside, when Martha asks what's happening, the Doctor, who is working on the machine, is holding several coloured wires, most prominently white and blue, in his left hand and has a white wire wrapped around his right hand. The scene cuts to Martha, and when it cuts back to the Doctor, the wire around his right hand is gone and he's only holding a white wire in his left hand. (00:29:10)
The Lazarus Experiment - S3-E6
Continuity mistake: When the Doctor and Martha get into Lazarus' machine, it's shown to be very cramped inside - when the Doctor crouches down to access the wiring and circuitry below a floor panel, his head is right next to the skirt of Martha's dress. However, after he tells Martha what he's doing to the capsule, when he notes that the transformed Lazarus is three times the size of his human form, the Doctor is further away from Martha. (00:29:25)
The Lazarus Experiment - S3-E6
Continuity mistake: After the Doctor and Martha get out of the capsule and walk up to the human Lazarus, lying on the floor, Martha says it's kind of pitiful, turning to look at the Doctor as she does so. In the next shot, facing the Doctor, Martha's head is still turned to look at Lazarus. (00:30:10)
The Lazarus Experiment - S3-E6
Continuity mistake: In the cathedral, when Lazarus says, "You're so sentimental, Doctor. Maybe you are older than you look," his blanket has jumped up to cover his left shoulder, which was previously bare. The blanket then changes position several more times. (00:34:50)
Continuity mistake: When the Doctor asks Captain McDonnell how many other crewmembers there are on board, he grabs her arms. After she tells him "three", the Doctor lets go of her as Scannell explains that the ship transports cargo across the galaxy. In the next shot, facing the Doctor, he's holding McDonnell's arms again. (00:03:00)
Continuity mistake: When the Doctor starts scanning Korwin, he's holding the sonic screwdriver with his thumb on the end without the light. When Ashton says that Korwin sabotaged the ship, the Doctor's grip on the screwdriver has changed to one without his thumb on the end. Shortly after, he has his thumb on the end again. (00:06:10)
Character mistake: The question to unlock one of the doors asks for the next number in the sequence 313, 331, 367. The Doctor gives the answer as 379 with a convoluted (albeit correct) explanation of happy primes, but there are much simpler solutions. The difference between the first and second numbers is 18 and the difference between the second and third is 36 (which is 2 x 18), suggesting that the difference between the third number and the next could be either 54 (= 3 x 18), giving 421 as the next number, or 72 (= 2 x 36), giving 439. It's likely that the Doctor would see that the answer is ambiguous, and yet he confidently states that the most complicated answer is correct. (00:08:45 - 00:09:15)
Suggested correction: If you know what a happy prime is, like the Doctor does, it might be simpler than your idea because you gave 2 options - 421 and 439. The idea of a pattern is there should be only one answer that fits, which would be the case if they're all happy primes.
Continuity mistake: When the Doctor comes up with the idea of using the generator to restart the ship's engines, in a close-up he's gesturing with his right hand, but in a wide shot is gesturing with his left. (00:10:15)
Continuity mistake: After McDonnell warns the crew that the possessed Korwin killed Abi, Ashton orders Erina to hurry up with the equipment she was sent to fetch, and she mutes the com before removing several pieces of equipment from a locker, holding them with her right hand and arm. When she closes the locker door, revealing Korwin, the equipment disappears. (00:15:20)
Continuity mistake: After Korwin cuts the power to the stasis pod, close-ups of the Doctor inside the pod show light still on in the pod. A shot from outside the pod, however, after McDonnell runs off to deal with Korwin, shows the inside as being dark. (00:35:25)
Continuity mistake: At the beginning, when the Doctor asks Martha if they saw her face, her hands change position, from grabbing his arms in a shot facing her to facing up in a shot with both of them. (00:00:10)
Continuity mistake: When the Doctor says, in response to Martha asking how the Family can follow them, "Stolen technology. They've got a Time Agent's vortex manipulator," in a close-up of his face he reaches with his left hand for a control on the console, but in a following shot of his hands, they're right next to each other. (00:00:25)
Continuity mistake: When Martha and Jenny are washing the floor, just as Hutchinson and Baines come along and tell them off for laughing, one of the lace ribbons on Martha's cap moves between hanging in the back in shots facing her and draped over her shoulder in shots facing the schoolboys. (00:04:15)
Continuity mistake: When Martha initially bursts into John Smith's study after hearing that he fell down the stairs, the door is mostly closed. When she runs back to the door to knock on it after Joan Redfern tells her off for bursting into a master's study unannounced, the door is open wider. (00:06:35)
Continuity mistake: During Joan's encounter with the Family's spaceship, when it shines a beam of green light at her, she shields herself by raising her right arm. It changes position between a wide shot and a close-up. (00:12:20)
Continuity mistake: After watching the Family's spaceship land, Jeremy Baines puts his hand on a tree stump as he walks around it to approach the ship. He does this twice, in two successive shots. (00:14:10)
Continuity mistake: When the Doctor fits the Chameleon Arch watch into the headset, his hands change position between shots. (00:20:15)
Revealing mistake: In the close up shot of the boys' machine guns firing, in the background a white truck can be seen driving across the green countryside; not appropriate for the time period of the story. (00:25:25)
Other mistake: Joan puts her stethoscope in her ears backwards when listening to John Smith's single heart.
Plot hole: The Doctor leaves Martha a recording with instructions for her for when he's human. However, the flashbacks to the incident that led to him using the Chameleon Arch don't leave enough - or any - time for him to have made the recording, presenting his decision to use the device as a spur-of-the-moment one that he immediately acts upon.
Suggested correction: He would have been planning the change to human for a while now, not just made it up on the fly. So he could have made a recording before the opening scene.
The episode makes it pretty clear that he and Martha got taken by surprise by the Family's attack. The Doctor also explicitly says, when he gets out the Chameleon Arch headset, that he never thought he'd use it. The flashbacks consistently present his decision to become human as one made as an urgent, in the moment one.
