In Fahrenheit 451, identify a problem and solution experienced by Guy Montag in the book.Looking for the problems Montag encounters and the ways he solves these problems.
Your question seems to refer greatly to the conflict within the book. One key area for me is the internal conflict Montag suffers through his thoughts about fire and his work, and how this changes as the novel progresses. At the very beginning of the book we see Montag's obsession and enjoyment of fire and its destruction of books:
It was a pleasure to burn.
It was a special pleasure to see things eaten, to see things blackened and changed. With the brass nozzle in his fists, with this great python spitting its venomous kerosene upon the world, the blood pounded i his head, and his hands were the hands of some amazing conductor playing all the symphonies of blazing and burning to bring down the tatters and charcoal ruins of history.
These first words of the novel express Montag's fascination and attraction to burning, interestingly linking it with power. It is clear that the metaphor used to describe his hands (to a conductor) shows that burning allows Montag to imagine himself transformed into some kind of artist - the power to orchestrate the destruction of "the tatters and charcoal ruins of history" clearly has its attractions to him.
However, the first burningwe are introduced to in the novel changes Montag's view and sparks off (no pun intended) the conflict that drives the rest of the novel and his own inner enlightenment. The woman who willingly burns herself, poignantly quoting Latimer, causes Montag to question the destruction of books. This also triggers off the battle for Montag's soul by Beatty and Faber who both attempt to win him over to their point of view.
After he has escaped the hound, he undergoes a transformation in his view of fire:
The fire was gone, then back again, like a winking eye. He stopped, afraid he might blow the fire out with a single breath. But the fire was there and he approached warily, from a long way off. It took the better part of fifteen minutes before he drew very close indeed to it, and then he stood looking at it from cover. That small motion, the white and red colour, a strange fire because it meant a different thing to him.
It was not burning. It was warming.
...He hadn't known fire could look this way. He had never thought in his life that it could give as well as take. Even its smell was different.
This passage marks the end of the conflict within Montag, when he realises the other properties of fire--to warm, nurture and protect, rather than simply to destroy. It is this that completes his process of enlightenment and prepares us for his decision at the end of the novel to join the group of dissidents and be part of the group that memorises texts rather than seeks to preserve the actual books themselves.
I think that you can consider the most elemental problem that Montag faces at the outset of the book. When Clarisse approaches him and asks if he is happy, it is a moment where Montag becomes acutely aware of who he is and the person he might need to be. It is at this moment that something becomes awakened within him, an element that causes him to develop a divided consciousness about his relationship with himself and his world. It is this question that triggers him to examine his life, his relationship with Mildred, and what he does. It is within this paradigm that the problem with how Montag's society is constructed and his role within it is addressed and it is at this point where change reveals itself.