This was a fantastic episode. There were two main plots:
The first was the case of the week, which involved mathematicians attempting to solve P vs. NP, a famous math problem that, if solved, would greatly affect the real world. Two mathematicians working on the equation were killed, and Sherlock and Joan are on the case to figure out what happened.
There were some good guest characters here, and some nice misdirection. Nothing so dazzling as to stick out of the texture of the average procedural, but it was a particularly good example of procedural television. I had trouble figuring out who the killer was, but in fact the obvious suspect was the killer - her alibi, which had earlier ruled her out, proved false. I like the way that Sherlock acts on these cases. Instead of just being a know-it-all, he really listens to and respects the opinions of Bell and Joan as he works through the case.
The real meat of this episode, however, took place in the other story line. Joan comes across the son of a former patient - the one that died under her care, and the reason she got out of medicine. This kid, played by Jeremy Jordan (of Smash fame) asks Joan to invest money in the bar he is opening. Joan asks Sherlock for a $5,000 advance, and Sherlock uses his famous deductive skills to figure out that this has something to do with her former patient.
One of the most brilliant things about this episode is what happens next. Joan tells Sherlock the story of what happened to the man, and how she messed up. She talks about how she got to know the wife and son, and while the man's wife sued her after she killed her husband, the son wrote her a letter telling her that he didn't blame her. However, since then he has solicited her for money a few times, and Sherlock sees how dangerous this is. Joan's guilt is obviously very strong, and it's preventing her from doing the right thing.
So, he gives her $20,000. He tells her to use it to buy the kid off for good and get him out of her life. Instead, Joan offers the money to the kid on one condition: he use it to go back to school. He says he'll think about it, but Joan is not optimistic. At the end of the episode, Sherlock says he'd like to go with Joan the next time she visits her patient's grave, since he knows how important of an influence he was.
Okay, so this is just perfect characterization all over the place. Sherlock doesn't quite know how to be a caring friend. I mean, offering someone $20,000 when they asked for a quarter of that and then telling them how to spend it... it's weird. It could be considered quite rude and controlling. But it's just Sherlock's way of showing he cares. And offering to visit the grave with her... that's him doing a better job of knowing what Joan needs. He has no personal reason to go to this man's grave, but he tells her that he does. He's doing all of that to meet Joan's emotional needs, and it shows how Sherlock has grown as a person since Joan came into his life.
This episode gets high marks from me. It had a very personal and very sweet story that spent most of its time focusing on the lovely Ms. Watson. And the case of the week was pretty strong as well.