Continuity mistake: The tear on Jeff's cheek briefly switches sides when he is holding the third card.
Continuity mistake: When the movie opens, Eric is in the bathroom and looks over at Adam, who is leaning up against the wall. However when we see Adam in Saw 2, he is propped up against the pipe. Eric hasn't moved him, as he's chained up. Obviously when they set up the bathroom for this movie, they did not put the dummy Adam in the same place. This scene takes place immediately after the scene in Saw 2, and thus there's no time for the body to have been moved at all. (00:00:50)
Continuity mistake: In the "pig" room the string around the tape player suddenly disappears in the middle of the tape message. Jeff's right hand never changes position and the string was tied around the middle of the player, he would have had to move his hand to get it off. (00:58:00)
Continuity mistake: When Jeff finds the second box (after the freezer room) he takes the card and flips it over to see the picture on the back. In the closeup his thumbs are on the bottom corners, but then the shot cuts away to a view from his left and you can see that his left thumb is at the top left corner and his right thumb is as least halfway up the card (instant cut, no time to move his hands). (00:48:30)
Continuity mistake: The blood stains (streams) on Jigsaw's chest constantly change after Jeff slices his neck. Sometimes the only blood stain is a trail that goes directly down the side of his hospital gown, other times it's halfway down his chest. The shots constantly go back and forth, so it's not just the blood flowing down his chest.
Continuity mistake: There is a smear of blood on Jeff's right temple that disappears in the shot right after he drops the saw blade. It reappears in the very next shot.
Continuity mistake: When Jeff flips over the 3rd card he is holding it with his left hand by the left edge of the card, but when the scene (instantly) cuts away to a view of in front of Jeff you can see the he's holding it by the bottom (look at the way the card it bent).
Answer: Adam didn't just lose because he let the key get drowned. He had the two hacksaws and he broke his own in a hurry. And there was also the toilet lid that he could have used to smash his foot like Eric did in Saw III. Jigsaw never said that improvising or thinking outside the box was against the rules. So even if you wipe the key out of the equation, Adam still has at least two other ways to release himself from the chains.
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