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Don't Eat the Snow in Hawaii - S1-E1
Continuity mistake: When Magnum drops the ring out of the envelope, nothing is near it as it's shaking. In the close up shot, it's next to keys. (00:44:20)
Continuity mistake: For the first episodes (minus the plot), the opening of the series features the Ferrari with two different license plates. The opening shot shows the ROBIN-1 vanity plate that is in every episode but not in the most well known version of the opening (with the iconic Mike Post theme music used in all the other seasons), and the following shows the license plate from the pilot.
Other mistake: Twice in the episode it is explicitly said that Magnum's phone has been disconnected, and that Higgins checked with the phone company. Yet at the end, it turns out 'the lads' chewed on the cable. Surely the phone company wouldn't tell the estate manager that the line has been disconnected for billing problems (so, a well determined administrative issue with proper paperwork done) when it's a wiring problem.
The Ugliest Dog in Hawaii - S1-E8
Continuity mistake: Inside his study when he talks with Magnum explaining the radiographs to him, Higgins is wearing trousers. Interrupted by the intercom, Higgins then goes outside with the intent to shoo off TC and the "juvenile delinquents", but does so wearing shorts. (00:12:40 - 00:14:20)
Revealing mistake: In the night shootout, Magnum cocks his pistol in a close-up that is obvious performed by a trained stunt double. Whoever is doing this is wearing the same clothes Magnum has but with the T-shirt in pristine condition straight from the washing machine, while Tom Selleck just took a dive in the dirt. Moreover, he wears the team ring upside-down. (00:22:50)
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Revealing mistake: In the opening of the second third of the season (the one with the iconic Mike Post theme) there is a brief scene with a car propelled in the air by an explosion. It's an outtake from the pilot movie, since the banged up car is very recognizable, but it's also easy to see why it was an outtake; you can easily see cables used to make the car fly in the air. Also, that car was not in an explosion in that episode.
Revealing mistake: In the final car chase, the military police radios for reinforcement; in the exterior shot that follows the one inside the limo with Rick on his back, foot on the cockpit, you can see quite distinctly the two stuntmen meant to be Magnum and Rick, and totally not looking like them. (00:46:30)
Continuity mistake: The plane at the beginning of the episode stops with the propeller at 7:05. When Louise DeBolt walks past the propeller to get in the cockpit, the propeller is at 11:25, and goes back to 7:05 next time it's shown. (00:02:10 - 00:03:10)
Thank Heaven for Little Girls and Big Ones Too - S1-E4
Plot hole: "Three Tahitians", one of the masterpieces from one of the most famous post-impressionist painters in the world, is authenticated by an ordinary school teacher. Because that's the person for this multimillionaire job, obviously. (00:25:20)
Visible crew/equipment: When Digger drives the Ferrari up to the gate of Robin's Nest, crew and location filming equipment are visible on the surface of the car. (00:09:45)
Continuity mistake: During the rescue escape, while Magnum is carrying Higgins on his back, Higgins has handcuffs around his wrists. Each of the locked wrist cuffs have a single and double bar, and during the span of the rescue these cuffs switch direction and face the opposite way around Higgins' wrists, which is impossible.
All Roads Lead to Floyd - S1-E13
Revealing mistake: While Magnum is a passenger in Floyd's car, in the closeups of both of them with Floyd at the steering wheel, there is black cloth over the door beside Floyd (presumably covering the device that secures the side camera to the door), which does not exist in wide shots.
All Roads Lead to Floyd - S1-E13
Visible crew/equipment: When Floyd is on the public telephone, the head of a crew member who is standing near him is reflected on the phone's chrome surface, though in wide shots nobody is near him.
Visible crew/equipment: After TC is taken away by the Marshals, it cuts to Magnum and Rick standing in front of the map with the Coast Guard personnel, and when Neil begins to explain where they intercepted TC, a white T-mark is visible on the floor. (00:18:35)
The Curse of the King Kamehameha Club - S1-E11
Visible crew/equipment: At the end, when Magnum tells Charlie that although it's all fascinating folklore it's tough to believe it. Charlie replies, "Maybe for a Haole," and in the next shot facing Magnum a part of the reflector screen, which is angled upward, is visible at the bottom, right corner of the screen. (00:48:35)
The Curse of the King Kamehameha Club - S1-E11
Visible crew/equipment: When the ambulance arrives and Rick's gurney is wheeled through the front doors, the reflections of the camera dolly and cameraman are visible on the glass. (00:35:45)
The Curse of the King Kamehameha Club - S1-E11
Visible crew/equipment: When Magnum is invited in to see Mr. Dollinger, just as Mrs. Macao tells the elderly gentleman to calm down and that he has a visitor, the two black tape marks are visible on the floor. (00:27:55)
The Curse of the King Kamehameha Club - S1-E11
Visible crew/equipment: While Magnum and TC are driving on the road to locate Makua, in the shots facing TC there's something attached to his passenger side door (for the side mounted camera). We can also see reflections of lighting, the side mounted camera (TC's side), etc, on both pairs of their sunglasses. (00:17:25)
The Curse of the King Kamehameha Club - S1-E11
Continuity mistake: While Magnum is at the morgue he ends up being interviewed and filmed by Christine. There's a closeup of the TV set which is broadcasting Magnum's interview, then it cuts to a wide shot of the TV set as Higgins clicks the TV off in annoyance. The two TV sets in the closeup and wide shot are completely different.
Visible crew/equipment: After Magnum gets Laura back, Newton's buddy drives backwards to leave and Magnum rushes off, then in the next shot of Newton leaning against TC's van, the reflections of crew and filming equipment are visible on the van's surface. (00:41:45)