“They battled the Pats hard at home the last week of the season...”
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And don’t forget, the Jints already had their play-off spot clinched. The outcome of that game meant nothing to their standing. They could very easily have just thrown their uniforms on the field and made sure no one would get hurt for the play-offs.
Oddly enough (in retrospect), the Patriots probably played that "meaningless" Week 17 game in the worst way possible. If they had obliterated the Giants, they probably would have demoralized them to the point where the Giants may not have had the confidence they needed to win four straight post-season games. If they had lost the game, the Giants may have just decided that being the only team to beat the Patriots was enough for a successful season.
In beating the Giants by such a close score, the Patriots probably gave them the right combination of confidence and motivation to carry them through the next four games.