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But it's a different 3D image on each screen, and playing games in which the display is split over 2 screens is what the DS and DSi are all about. You view both screens with both eyes, not left screen for left eye/right screen for right eye.
Could you not see the 3D effect in that video? It's nothing like "magic eye" pictures" whatsoever. Rotate the handheld and you see behind the objects that are intended to be at the front of the display just as in real life you move your head to look around the sides of a object.