Is there an elegant mathematical proof to assert that we cannot see more than 3 faces of an opaque solid cube simultaneously (of course without mirrors or any optical tools such as camera, etc)?
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There are three pairs of opposite faces. To see more than three faces, you would have to view both from one pair simultaneously.
Opposite faces of the cube are parallel. If you can see a particular face, you can't see the one opposite it, because look which side of it you're on.