Continuity mistake: When Radar walks into Henry's office to tell him that there's a general there to see him, we get a good view of Henry's new desk and its beautiful ornate design, but later when Hawkeye and Trapper are carrying the desk and then it's lifted out by chopper it's a completely different desk - also note that this desk has casters.
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Visible crew/equipment: In the opening scene where General Clayton is explaining the 4077 to the psychiatrist, the scene changes to the outdoor set and the camera pans to the right. As the camera pans past the hospital, a white 1970s era shuttle van can be seen driving into the set in the upper right corner of the screen.
Continuity mistake: The "KILROY" Hawkeye writes on the window as BJ looks out was in the center with his fingertip. Later, as Hawkeye looks out, the "KILROY" is now closer to the frame, and the lettering is different, a small capital A in front of it, and lines above and below it.
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Plot hole: When Hawkeye wrecks his jeep on his way back from an aid station, he has a Korean girl take a message to the 4077th, which he claims is 20 miles south. However, in virtually every episode, it is claimed that the 4077th is located 4 miles from the front lines.
Plot hole: When the truck with the orphans arrives, Col. Potter and Nurse Cratty greet each other like one would an old friend. Being new to Korea, Col. Potter doesn't know Nurse Cratty - as becomes clear from his reaction to Hawkeye's explanation to BJ earlier (who doesn't know Nurse Cratty for the same reason, but that's not relevant for this mistake). Note that in other parts of the series Col. Potter is depicted mostly as somewhat old-fashioned and reserved towards strangers.
Plot hole: When Hawkeye and BJ buy Margaret a new ring, it takes a typo in the inscription to give them away - which makes both of them and Mr. Shin almost clairvoyant. Not only did they somehow guess Margaret's ring size, but Mr Shin - phew, Houdini and Uri Geller together don't hold a candle to this guy - actually somehow divined the exact type and font of the inscription in the original so perfectly that Margaret actually only spotted the difference from the missing 'n.'
Continuity mistake: When Cpl. Hendrix is examined on the operating table, Klinger and Hawkeye admire his tattoos. Klinger asks "do you really think the real Louise is built like that?" and Hawkeye replies, "If she is, I'm in love". Later, when Radar talks to Hendrix, Radar says, "Her face is still pretty, and her legs are the greatest." Hendrix asks, "What about the rest of her?" and Radar replies, "There isn't any." How could Hawkeye and Klinger admire Louise's build if the tattoo was that shot up?
Factual error: Nearing the end of the show, we are told that the Chinese have become involved in the conflict and have attacked with some 300,000 soldiers. This apparently refers to massive attacks by the Chinese starting on November 1, 1950. Although the numbers are correct, the US/UN forces were unaware of the numbers. As late as November 6, the Far East Command continued to insist that there were no more than 34,500 Chinese communist soldiers in country. http://www.history.army.mil/brochures/kw-chinter/chinter.htm.