Revealing mistake: Thornhill & the "Professor" have a chat on the airport tarmack, in which it's revealed to Thornhill that Eve is a secret agent and in danger. At that moment the camera dollies in on Thornhill for dramatic effect. As it does, the distant airplane in the background gets closer very nearly as fast as does actor Cary Grant, revealing that the background in this scene is, in fact, a rear projection on a screen just beyond the actor. (01:40:15)
Revealing mistake: Right after Eve "shoots" Roger and meets up with him and The Professor, some of the trees wobble as the ambulance drives past them. (01:46:20)
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Revealing mistake: In the shooting scene in the Mount Rushmore cafe, a boy in the background puts his fingers in his ears, because he knows the gun is about to be shot.
Revealing mistake: When Thornhill is going up the driveway of the house near Mount Rushmore, there are shadows, even though it is night-time! These shadows are far too strong to be moon shadows and there is no source of artificial light that could be making them.
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Revealing mistake: When Thornhill's cab arrives at the UN there's a couple behind him, a rear projection. Later on, when Valerian's car arrives at the UN the same projection is used, with the couple a mere two meters away form where they were before.
Revealing mistake: Just before Roger Thornhill exits Eve Kendall's train compartment, they are kissing and have their hands cradling each other's heads. When Roger turns his back to the camera and Eve's hands are behind his head you can see a tan line where Eva Marie Saint's wedding ring usually is. Her character, Eve, is single.
Answer: Not really. You could (and at some hotels are still able to) keep your room number private or you could not - i.e. you could ask the hotel staff to keep your number secret from strangers, or you could ask them to tell anyone who might ask. Not having seen this movie, I don't know how likely it would be in the situations you speak of that the hotel guest would choose the latter option- it might be a mistake.
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