Continuity mistake: First, Scotty rigged a self destruct delay timer for Kirk that supposedly had thirty seconds programmed in. However, putting a timer against the one used in the episode, from start to finish there was one minute twenty seconds before the Constellation blew up. Second, Kirk was holding the communicator out and away from himself before his beaming, then it's close to his face as he materializes.
Continuity mistake: Decker is leaning over the forward console between Sulu and the other man, with Spock at his left elbow. As he gives the order to veer off, he starts to turn to his right. As the angle changes, it appears that he was standing straight up the whole time.
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Revealing mistake: While Kirk watches, the machine fires at the Enterprise. As Spock is knocked backwards, watch his hands as he goes forward. They automatically go into a certain, almost practiced position. Then a moment later, the Communications officer is reporting to Spock. As she does, another shot hits the ship. She and just about all the crew react to the jolt a second before the camera and lighting crews simulate the hit.
Other mistake: When preparing to go around again, Sulu and Checkov start working their controls. When there is a close up of Sulu's hand, it's obviously a re-used shot of Kirk's hand starting the 30 second count down timer on the Constellation in "The Doomsday Machine" (S2 E6).
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Revealing mistake: In the tent after Kirk and friends give up their weapons, there is a small fire in the holder in the center of the room. Although the flames are slightly off center to the camera's left, the are projections of flame light from off the camera's right offscreen.
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Other mistake: When talking to Edith Keeler one evening, Kirk refers to a star in Orion's belt, pointing it out. The shot then cuts to a scene of the horizon. The sky is quite cloudy and the light from the sun is still lighting the horizon. The stars would not be visible.
Continuity mistake: Kirk has a sunburn on his face and chest (it appears to end partway down his chest-most noticeable in the scene shortly after his identity is revealed to the Klingons). First of all, how does one get a sunburn while flying through space (and one that does not match the pattern of the uniform). If one wants to explain it away as being from a recent shore leave or Enterprise tanning bed, the issue still remains that the intensity of the sunburn changes as the episode continues.
Deliberate mistake: When the Klingons post signs for their rules, they're printed in English. The Klingons have their own language, and people on Organia most certainly aren't English speakers. Even if the Universal Translator allows Kirk and Spock to hear in English, it wouldn't change the look of a sign.
Continuity mistake: As Kirk is crawling through the tunnel leading to the Horta's nest, a tight view from his left side shows light shining on his face and darkness behind him. When the shot changes, though, to the view from the nest, there is light behind him and darkness on his face.
Continuity mistake: While Kirk is attempting to convince Marplon and Reger to take them to Landru, a distant shot shows Marplon's right hand descend from his chest to his side. Immediately thereafter, a close-up shows Marplon's hand again suddenly at his chest.