Other mistake: After Jeff gets the key from the Rack trap room, the shotgun fires and hits the judge. He is standing with the shotgun to his right, yet he is hit on the left side of his face. There is also an inconsistency with the timing. The gun fires, then the actor turns to reveal the wound on the left side of his face. It's easier (but in no way required) to see the glaring mistake with slow motion.
Other mistake: Near the end of the film when Jigsaw has the tape player, the 'play' button is near the left side. But Jigsaw's finger is on the right side, not even close to the play button. Jigsaw presses on the right side where there isn't even a button, but the tape begins to play anyway although nobody pressed the actual play button.
Answer: It was too late for Adam to solve the game the way he was intended to- the key went down the drain, which Jigsaw actually didn't intend to happen. Adam's key was meant to be tied around his neck in such a way that neither him nor Lawrence would be able to see, and Adam would be allowed to go free and escape if he were to locate the key- which would only be possible if he looked at himself. The whole idea behind his puzzle was "looking at himself instead of others, for a change." The reason it went down the drain instead, was because of Amanda, who continuously made traps unsolvable. Instead of tying it around him like Jigsaw asked, she just tossed it on his chest, which defeated the whole purpose. Also, he was given a saw just as Lawrence was, as a much more violent plan B, another way he could save his own life.
That doesn't make sense because Jigsaw told Adam that the key is in the bathtub at the end of the movie.