Deliberate mistake: The credits list "Number 86" as played by John Maxim. But no such character appears in the episode. Apparently, 86 was left on the cutting room floor, but the titles weren't altered accordingly.
Continuity mistake: When No.2 is writing on the blackboard, in Question A. The capital letter "F" changes several times between the long shots and close ups. Also in Question B. The position of the first "T" in the word together also shifts position during the same shots, showing that the scene was filmed twice.
Continuity mistake: In the court room scene, Number 6 has wind-blown hair in the master shot, but is nicely combed up and neat in all the close-ups.
Revealing mistake: When they're "playing" on the swing, Number 6 has his hands on the Butler's shoulders, and the Butler is wearing gloves - yet there's a close-up of someone's bare hand holding the rope.
Continuity mistake: When Number 6 is pushing him on the swing, the Butler's hands and arms change positions instantly between shots. One moment, he's holding the rope with his hands: in the next, he has his elbows crooked around the ropes instead.
Continuity mistake: The sling that's been around the Butler's arm and shoulder vanishes after he fans Number 2 with a towel. It's back a few shots later.
Continuity mistake: During the interview, the Butler puts a coffee pitcher down beside Number 6. In the next shot, it has disappeared.
Continuity mistake: While he's making his graduation speech and demanding to know why Number 6 resigned, Number 2 has a mortarboard tassel wardrobe malfunction: it keeps changing sides between shots.
Chosen answer: We're led to believe Rover is some kind of robot in the series. According to books on the show, Patrick McGoohan was looking for an unusual guardian for the Village, looked up at the sky, and saw a weather balloon, which was what Rover was.
Captain Defenestrator