A man is being confronted by a mugger. He insists that he does want any trouble, only for the mugger to punch him out. The mugger slips into a driveway to count his money, when he hears gunshots from the house. The mugger repeats what the victim told him, only to be shot twice. Sherlock arrives at the case, and explains to Bell that he got a text message from him. When Bell denies such a thing occurring, Sherlock checks his messages again and finds out it was someone else. Already at the scene, Bell allows him to have a look. The first victim was a mathematician. Sherlock and Bell search the house for clues, when Sherlock notices there is an empty room. Sherlock darkens the room, and shines a black light, revealing math formulas written on the walls.
Joan is at the grave site of the patient who died on her, meeting with his son Joey Castoro. They go for coffee and Joey asks her for money. He intends to open up a bar and needs funding. Back at the Brownstone, Sherlock has hired a mathematician to see if he recognizes the formula. It turns out to be the famous P versus NP, a mathematical formula that has been pursued for years. While the formula is not complete, the discovered research is the closest anyone has gotten to solving it. The mathematician points out that the writing was done by two people, and also suggests he visit Tanya Barrett, a professor who wrote a paper about P versus NP. Watson asks Sherlock for a raise to give Joey the money, but Sherlock thinks that Joey is guilt tripping her into doing so.
Sherlock and Watson visit Tanya at the university she teaches at, and she points out a potential suspect, the second mathematician. She claims the two had been arguing and recently had a large fallout. Sherlock calls Bell to tell him about the suspect, but before he has a chance to say so, Bell informs him about another victim, naturally the same man. Sherlock and Watson visits the man's apartment, and Sherlock notices something strange about the security cameras outside. He hails a cab, stands on the roof, and notices a device on the camera pointing into the door. Sherlock goes to the man responsible for the surveillance: Linus Roe. When questioned, Roe states the P versus NP was going to bring a huge change in security. With it any password could be hacked. Sherlock replies to Roe that he must have wanted to get ahead and prepare for this new era. However, Roe actually wanted the mathematicians to solve it so his security company could exclusively 'P versus NP'-proof against it before anyone else.
The police bring in Tanya for questioning, and claim the bullets and dog hairs from the crime scenes match the gun and dog she owns. Tanya agrees, but points out that the gun was reported stolen and she has an alibi, going out for drinks with a friend. She says the gun was stolen by her ex-boyfriend Jason Harrison, who had been harassing her with emails. The restaurant sends them the footage of the times she claims she was present, and her alibi appears to be true. They bring in Harrison, who denies involvement, claiming to not even know the victims. Gregson points out that he sent threatening emails to Tanya and claimed he complained that she cared more about the mathematicians than him.
Just as Harrison seems to be the culprit, the mugger wakes up and is questioned. He identifies his shooter as Tanya Barrett. Sherlock and Watson look over the footage again and Watson remarks how cheap the beer is given how much is being sold. Sherlock realizes the video was shot during happy hour, which is not the time the video seems it was shot at. Sherlock and the police confront Tanya, tell her that she used the P versus NP code to hack the time-stamps and send the emails, and Sherlock informs her that her partner confessed. Sherlock and Watson go back to the Brownstone and he agrees to give her twenty-thousand dollars to get rid of Joey once and for all. Watson offers the money to Joey, but only if Joey agrees to go back to college like his father wanted him to.
Tropes:
- Asshole Victim: A mugger who just punched out a man stumbles onto a murder while running away, and gets shot twice in the chest for his trouble. He doesn't die, and him waking up from a coma is why the killer is identified.
- Carry the One: When approaching a math professor scribbling away, Holmes makes this joke.
- Frame-Up: Tanya uses her hacking program to send incriminating emails from her ex-boyfriend's account, giving him a motive for the murder of the mathematicians.
- Ironic Echo: A mugger punches out his victim when he says, "Please, I don't want any trouble." Said mugger stumbles onto a murder while running away, then says the same thing to the murderer before getting two in the chest for his trouble.
- Shirtless Scene: Sherlock is shirtless and doing exercises in the middle of the night in the brownstone.
- Suspect Existence Failure: Holmes calls Bell to tell him who he thinks the killer is, and before he has a chance to say so, Bell informs him about another victim, naturally the same man.
- The Unreveal: We never get to hear the solution to P versus NP, although since it hasn't actually been solved in real life, that was pretty much a Foregone Conclusion.
- Writers Cannot Do Math: For a math-focused episode, there is quite a lot of artistic license:
- The scribbles on the wall contain some math that is very unlikely to be used in a proof of P vs NP.
- Holmes' associate Harlan Emple claims that the victims were very close to solving it, a statement that is impossible to make unless one knows the solution already. Properly done, this could have been used as another clue that Tanya was the perpetrator, as she makes the same claim.
- There are three possible solutions to the P-vs-NP problem: They are equal; P is a proper subset of NP; the statement is unprovable. Only the first one could be a danger to much of modern encryption, and even then only if there are efficient polynomial algorithms. A proof that P=NP doesn't necessarily provide a polynomial algorithm to solve any NP-complete problem either and may only prove the existence of one.
