I believe the main conflict is Auggie dealing with bullying from the other students because of his physical appearance.
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The barber Ivan Jakovlevitch finds Major Kovaloff's nose in his bread. It happens all of a sudden, without any introduction, on a usual morning. What makes this occurrence even weirder is the fact that he recognizes the nose, as if all the noses in this world aren't similar, and instantly starts quarrelling with his wife, who accuses him of having chopped off a customer's nose.