Plot hole: Solo blows open the ceiling vent of Illya and Marion's cell, and she stands on Illya's shoulders until Solo can reach her and pull her out. This leaves Illya with nothing/no one to stand on, no way to reach the vent and no apparent way out of the cell. But he's out just the same in the next scene, with no explanation as to how.
Factual error: Ole and all the Thrush baddies call the missing chemical component a "catalyzer," repeating the term throughout the episode. This was a scriptwriter's error, which only David McCallum, to his credit, corrected: he had Illya say "catalyst," which is what the writer meant. "Catalyzer" isn't a word.
Continuity mistake: When Solo begins trying to seduce the dancing girl with the line, "We shall taste paradise," he's standing at least a foot away from her. But in the very next shot, they're suddenly in a clinch: her hands are on his cheeks and they're about to kiss.
Plot hole: U.N.C.L.E.'s pen communicators worked much like modern cell phones: when one agent called another, they "rang" with a 2-tone signal. For some reason in this episode, Solo's pen, which isn't on an open circuit, transmits Illya's voice without ringing first. It signals normally a few scenes later, though.
Continuity mistake: Solo picks up a book, The History of Pacifism, and shows it to Illya, holding it at the bottom. Cut to a close-up insert and his fingers have instantly migrated up to the middle of the book. Cut back to the wider shot, and he's no longer holding the book at all. It's back on the table.
Factual error: The jailed student protestors are dancing in their cell to a jazzy tune. But they have no radio or record player - just one guy with a guitar. They seem to have conjured the accompanying saxophone, jazz clarinets and percussion section out of thin air.
Factual error: Illya leaps onto the rope in the monastery tower, causing the bell to ring. After sliding down the rope to the floor, he then identifies the tone as "B Major 7th." But the musically-literate Illya should know better. B Major 7th is a chord (B, D-sharp, F-sharp, A-sharp), not a single note.
Continuity mistake: At the end, Solo and Illya are sitting at a small bistro table as the captured bad guys are marched past them. Solo wears a green fez, and between shots, Illya suddenly acquires a red one. It wasn't in his hands or on the table before that: it just seems to appear on his head out of nowhere.