During the fight of Kirk and Spock versus Chekov and two security guards, Spock renders one guard unconscious with the Vulcan Nerve Pinch. However, about 10 seconds later, Kirk has Spock escort all three to the brig, all of them walking, including the guard that had been nerve-pinched.
In the Transporter Room, Spock states he can't beam the security detail back because he couldn't lock onto the proper coordinates. Then they find out the Enterprise is not orbiting the planet. How were the guards beamed out without accurate coordinates?
After Kirk and Spock leave Tommy on the bridge, Tommy looks up toward the main view screen. We had seen Sulu and Chekov at their posts soon before and again soon after, but, in between, a stock shot is used with Hadley at the navigation station instead of Chekov.
Uhura is distracted from contacting Starfleet by seeing herself in her mirror as a very old woman. However, in all of the wide shots, there is no mirror.
When the children summon the angel on the bridge they start the chant with "Hi, Hi, fire and snow" When Captain Kirk orders Spock to play back the recording the chant starts with "Hail, Hail, fire and snow".
The transporter was used to send a security detail down to the planet. The red shirt controlling the transporter was not under the control of the children, so he would have noticed that the Enterprise was not in orbit before beaming the two security red shirts in to outer space.
Spock and Capt. Kirk are unable to use the auxiliary controls because Mr. Scott and two of his engineers won't let them, and Spock and Kirk are out-manned. Why don't Spock and Kirk return with phasers and stun the engineers? Better yet, why don't they stun the child who is controlling the engineers?
Even if Mr. Sulu believes that the Enterprise is still orbiting the planet, it's very unlikely that the other 429 members of the crew, especially the captain and Mr. Spock, wouldn't have noticed the ship jumping to warp, as the engines make a very specific and noticeable sound when the ship enters warp. The Enterprise has windows, so someone must have seen stars passing rather than the planet they're supposedly orbiting.
When Tommy first asks Kirk to take the children to Marcus-12, Kirk replies that it is not on their patrol route, implying that the Enterprise wouldn't be going anywhere near it.
Yet at the end of the episode, Sulu announces "Marcus-12 dead ahead."
The stardates the Enterprise and Professor Starn's scientific team used apparently do not match up. The opening stardate of 5049.5 would have made more sense than the 5029.5 quoted on the captain's log.
After the credits, Captain Kirk's log states that they've buried the bodies of the colonists and we see the graves but none of the members of the landing party have dirt on their uniforms. Nor are they sweating, as one would after digging human-sized holes in the ground.
The children's chant to call the "Friendly Angel" starts at first with "Hail, hail" then later with "Hi hi."
The children sometimes say Marcos-12 and sometimes Marcus-12.