Why does he run around the house in order to know on the front door to arrive after Daisy?
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Gatsby doesn't want to reveal to Daisy how much he has done to try to be an appropriate husband for her when they meet again. He is also very nervous about finally seeing her again after so long. A "proper gentleman" wouldn't reveal his emotions and Gatsby has worked diligently to promote his image as the ultimate gentleman in order to be worthy of Daisy. Thus he runs around the house so that he can arrive after Daisy and mislead her in to thinking that he isn't fazed by finally meeting up with her again.
The Great Gatsby