As you may remember, Watson stopped being a surgeon because a patient died under her care and she got sued for it. While she's out at the patient's grave, his son asks her out for coffee. He's dropped out of college, but he's got a great plan to buy a bar and all he needs is five thousand dollars. This is obviously a scam and the kid is just going to blow the money, but Watson still spends most of the episode waffling and asking Holmes for the money. At the end, she offers to pay four times as much for the kid to go back to college, but he doesn't go for it. Still, Holmes gets her to try to consider her mistake to be an accident that's best left in the past, which is nice.
Meanwhile, a mathematician has been murdered. And a mugger out in the alley has been shot, although no one thinks he's important. Holmes quickly reveals that what appears to be an empty room actually has walls covered in writing only visible under a blacklight. This writing is the kind of crazy-person mathematical doodling that you might remember from A Beautiful Mind, and it has to do with the "P versus NP" problem. In this show, solving that problem will have two immediate results: you'll win a million dollars and you'll also be able to crack any computer security or encryption in the world. So you can see how it's suspicious that someone who was working on that has been shot. Judging by the handwriting on the wall (har har) he had a partner, but no sooner does Holmes identify that person than that person turns out to have also been murdered. Oops!
The second dead person was being monitored by a security company that was concerned about the prospect of computer security suddenly being obsolete. But they had been assured that his solution was nowhere close to being done. And who assured them of that? Professor Tanya Barrett, who also helped out identify the partner in the first place. So she's obviously suspect number one, especially since her dog matches hairs found at the scene and she had a gun that matches the shootings. But she has an ironclad alibi, as shown by a restaurant's security footage and its embedded timecode. So maybe she was being framed by her ex-boyfriend, who sent threatening emails!
And then the mugger wakes up from his coma and pins the murder on Barrett. It turns out she had the solution to P versus NP all along, and she used her magical security-breaking powers to hack her boyfriend's emails and change the restaurant's timecode. She also stole hundreds of millions of dollars, but they don't really figure into the plot.
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New York at night. A mugger waves a knife in a victim's face. After he gets the guy's wallet, he ignores the protest of "Please. I don't want any trouble" and punches him in the face. We follow the mugger into an alley, where he sees a shooting through a window. When the shooter comes down into the alley, the mugger tries saying he doesn't want any trouble. And it doesn't work this time, either; the mugger gets shot.
A graveyard. A woman lays flowers at the tombstone of Gerald Castoro, who was a beloved husband and father. Oh, hey! It's Watson! And she's noticed by someone named Joey who calls her "Dr. Watson." Joey strikes me as someone who's up to no good, but to be fair, that's just because he's played by Jeremy Jordan, who played Jimmy the Jerk on Smash. It's probably not fair to hold that role against him, but that's what's happening. Deal with it. She's putting flowers on Joey's father's grave, and of course that's what he's here for. He came the day after the anniversary to avoid his mother. Watson agrees that he was a good man. Gerald lays down his flowers and offers to buy her coffee. She tries to beg off, but he insists.
Gregson and Bell are at the alley crime scene where the mugger was shot. Holmes strolls up, apologizing for being late, and Bell says that he's not actually wanted at all, since he just heard about the murder on his scanner and Bell texted him not to come. Holmes claims to have received a text saying, "Yes, please. Now. Triple smiley face, with tongue protruding." Bell points out that Holmes is actually reading a text from someone named "Bella" from months ago. And with that out of the way, we might as well let Holmes work the case, right?
The victim inside the house is Felix Soto, a math tutor who lived alone. He's lying on his back in the middle of the room. The mugger's name is Benny Charles, and he seems not to have known Felix. They know that he was mugging someone right before, so he seems to have just ducked into the alley to count his swag. Which we, the audience, happen to know is exactly what happened! Benny survived, but he's in a coma right now. He's just waiting until a plot-relevant time to return to consciousness.
Holmes looks around and decides there are too many remaining valuables to assume a motive of robbery. Bell says a neighbor heard an argument. Holmes goes into an empty room and investigates the walls from an inch away. Bell noted on his own that it's weird that just one room would be completely empty. He considered the possibility that the shooter took something off the walls, but there aren't any nail holes. Holmes asks if Bell has always been this observant, and Bell gets in some quite acceptable sarcasm about how before Holmes came along, no one in the NYPD had ever closed a case.