Continuity mistake: Kirk and the others are beamed down to the planet, and when the Enterprise crew are led into Mudd's throne room by two Alice androids (the Alice series are 500 identical androids) they are wearing the usual metallic, backless, kitten heel slip-ons, but in the next shot android Alice at the left is now wearing altogether different shoes - cream color with ankle-straps. (00:09:30)
Visible crew/equipment: A stagehand's foot is visible before the party beams down. (00:09:10)
Continuity mistake: In Norman's chamber, the medallions on Alice 3 and 11 go from blinking to steady, to totally off and back to blinking through the scene.
Other mistake: When Norman goes to the second panel in Auxiliary Control, there is clicking when he turns his right hand, but there is no knob, just a blinking light.
Plot hole: How did the android Norman get aboard the enterprise? If he beamed aboard I'm sure someone would have noticed and where did he beam from? The Enterprise was nowhere near any planet and I'm sure they would have detected any spacecraft nearby.
Suggested correction: These are unanswered questions, not plot holes.
But something phrased as a question because it has no decent answer can constitute a plot hole.
