What is the problem and the solution in Bud, Not Buddy, by Christopher Paul Curtis?

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In Curtis' novel, Bud, Not Buddy, the problem is that Bud has lost his mother—she died when he was six. He has been in the Home for several years, in and out of foster homes, has been beaten up, and has learned to keep the truth to himself if he can, while keeping a sense of humor when possible. The problem is also that in the midst of this sad life he now leads after his mother's death, that he believes he holds clues to the identity of his father. He has to find a way to get to his father. The plot moves forward as Bud does all he can to come up with the means to get to Grand Rapids, Michigan, from Flint—to track his father down. 

In the library, Bud borrows the atlas to figure out the distance between Flint and Grand Rapids, Michigan. 

I went over to the table and found Flint and Grand Rapids in the lines of the book. I looked where the two lines met and it said 120. Wow! That was going to be a good little walk.

Bud decides he will travel to Grand Rapids, walking if he must, to find his father.

Part of the solution comes in the form of Mr. Lefty Lewis who spots Bud walking at night along the side of the road. Knowing that this is not a safe place for a young black boy to be (alone in the country at night), he takes him to his daughter's for food, better clothes and a good night's sleep. Part of the solution to the problem presents itself when Mr. Lewis agrees to take Bud to Grand Rapids, believing he has run away from home. Once he arrives in Grand Rapids, Bud is in a position to track his father down and figure out a way to stay there and not have to return to the Home. Mr. Lewis explains:

I got in touch with your daddy to let him know you were all right...I didn't call him, I sent a telegram...

True to his word, Mr. Lewis drops Bud off at The Log Cabin where his "father," Herman E. Calloway and his band are performing.

Once there, Miss Thomas and the band members are a great help. They let Bud tell his story. He reveals his mother's identity...

Angela, sir...Her name is Angela Janet Caldwell.

Then he shows them his possessions (most especially the labeled rocks) and everyone realizes that Herman E. Calloway is his grandfather. This transition is not easy for anyone, but Bud finds a place where he feels he belongs: a home with his grandfather and the members of the band.

It is with the help of all of these people that Bud finds his way "home"—no longer needing to travel. This is the solution to his problem.

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