Deliberate mistake: The scenes involving the 8mm projector and screen did not also contain legitimate light rays from the projector to screen or show any obstructions on the screen despite Joe or someone else between the projector and screen. What showed on the screen is what would be expected if using a videotape or DVD.
Other mistake: Joe and Alice were somehow able to get the sheriff down from his inverse suspension, but - given the height of the ceiling and depth of the hole - the sheriff's feet (where they would be tied for the inverted suspension) would be out of reach. Moreover, surely a knife or cutting instrument of some kind would have been needed to cut the rope (assuming a rope was used) - did Joe happen to have something in his backpack? What was shown to the viewer was Joe and Alice using their hands on the sheriff's waist to lightly sway him two or three times in an attempt to get him down. The viewer next saw the sheriff when all but his feet had landed on the ground. For the sheriff to fall unharmed - head first - is far-fetched.
Other mistake: When Joe picks up one of the cubes and puts it in his jacket pocket, Charles is nowhere around him. Later, Charles tells Joe that he told him not to touch it [cube] in the first place.
Answer: The explanation for the police car going into the auto dealership when Joe and friends are at the dinner is provided by J.J. Abrams in the "Commentary": "As it was originally written, you [audience] were simultaneously with the father going into the car dealership... umm... and with the kids in the diner. We ended up moving the structure around. So, it wasn't... umm. It was no longer simultaneous." Abrams said that the kids had been rehearsing at the table. Filming the auto dealership scene was finished so they went ahead and filmed the kids at the diner, too.
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