What are some examples from To Kill a Mockingbird that demonstrate the theme of using reason and intelligence to solve problems?

Because Maycomb is a town rooted in tradition and superstitions, Lee satirizes its idiosyncrasies and creates characters who separate themselves from others because of their use of logic.  Here are several specific examples:

1. When Heck Tate tells Atticus that Tom Robinson is being moved to the Maycomb jail and warns him about the mob that might be stopping by, Atticus calmly grabs a few items from the house and goes over to guard the jail.  He does not allow emotion or panic to overcome him; rather, he believes that he can reason with whoever shows up (Scout actually ends up solving the problem for him), but Atticus's presence at the jail no doubt comforted Tom.

2. Judge Taylor wisely asks Atticus to defend Tom Robinson.  He knows that the trial has the potential to be a riotous disaster, but with Atticus as the defense attorney, the judge is certain that that is the best solution to a potentially volatile situation.

3. When Scout emotionally tells Atticus of her horrid first day of school and of Miss Caroline's demanding that she not read at home anymore (simply an unintelligent order from a teacher!), Atticus suggests that Scout continue reading and not say anything at school about it. He has a way of calming down other characters' emotions so that they can see a more sensible approach to their problems.

4. Atticus needs to prove that Tom Robinson did not attack Mayella; so he uses all of his observations (the location of Mayella's injuries, Mr. Ewell's being lefthanded, and Tom Robinson's permanent arm injury) and builds his case from practical, irrefutable evidence.

5. When Calpurnia realizes that a rabid dog is near the house and the children, she reacts quickly and precisely.  She brings the children into the house, calls Atticus, and even risks her life to run to a neighbor's house to warn of the danger.  During all of this, she is able to negate the telephone operator's silly superstition about rabid dogs and accomplish what she needs to in order to keep everybody safe.

6. Dolphus Raymond knows that the townspeople would not leave him alone if they knew that he willingly and sanely married an African American woman and fathered children with her.  So, instead of demonstrating his reason and intelligence to everyone, he pretends to be drunk, all while getting to live the lifestyle he wants without being criticized.