Due to conflicts or challenges, a dynamic character is one who changes (either in personality, character, or attitude) as the story progresses.
In A Problem, Sasha has forged his uncle's signature on a false promissory note. When he can't come up with the cash to cover the fifteen hundred roubles on the note, Sasha's two paternal uncles (one a Colonel and the other, an official in the Treasury) and maternal uncle (Ivan Markovitch) hurriedly call a family meeting.
The Colonel is the most vocal of the three uncles in arguing for Sasha to accept whatever punishment is meted out for his crime. To fail to hold the young man to account would be an act of 'civil cowardice,' asserts the Colonel.
On the other hand, the uncle who works at the Treasury is only interested in preserving the reputation of the family; he is especially concerned that the distinguished family do not suffer any negative publicity from Sasha's disgraceful conduct. Ivan Markovitch, the maternal uncle, leans toward showing mercy; he reasons that his nephew has had a difficult childhood, bereft of 'good, benevolent influences' since he lost his parents at an early age. Therefore, Markovitch argues that Sasha deserves every 'indulgence' and the 'sympathy of all compassionate souls.'
The major part of the short story concerns opposing arguments between the Colonel and Markovitch regarding the proper cause of action for dealing with Sasha's deplorable conduct.
As for Sasha, his initial reaction to the whole mess is an apathetic one. He's not to worried about his family's honor either. The author states that Sasha feels neither 'terror, shame, nor depression' for his actions but only 'weariness and inward emptiness.' Sasha is not so much sorry that he is in such a difficult situation, he is merely sorry that his actions might just lead him to lose his ability to indulge in his carousing lifestyle.
Interestingly, Sasha is less apathetic about his thoughts regarding his Colonel uncle. He is quite furious that the Colonel has portrayed him as a criminal who must be punished. It appears that the Colonel wins the first round of debate, but it is a pyrrhic victory (a victory won at too great a cost). Even the idea of letting the case go to trial depresses the eloquent Colonel. Subsequently, Markovitch's pleadings for mercy on behalf of Sasha's deceased mother works its magic, and the Colonel angrily relents.
So, all three uncles argue for the best solution to the problem, but one uncle manages to best the other two. Markovitch is ecstatic that he has managed to win mercy for Sasha. However, his victory is a hollow one. When Sasha finds out that he is to go scot-free for his crime, he physically intimidates Markovitch into lending him a hundred roubles.
Sasha has now progressed from being sullenly apathetic to being overtly contemptuous of the misplaced mercy that would spare him the rightful consequences for his actions. He doesn't even bother to preserve his initial outward (albeit false) manifestation of remorse; his immediate act of extorting money from the uncle who just fought for his 'rights' proves that he has now become emboldened in his ways. In this way, the vacillation and inconsistency of his uncles have contributed to a negative dynamic change in Sasha.