- When a well-respected mathematician is found dead, Holmes and Watson set out to solve a murder and the purpose of the highly guarded equation he was trying to decipher.
- "Elementary" - "Solve for X" - October 3, 2013
A man is accosted by a mugger with a knife. The mugger whacks him on the head as he takes off with the man's wallet. The mugger hears a gun fired in an apartment, and then is shot by the shooter escaping.
Joan visits a grave and runs into Joey, the son of the dead man. He died while being treated by Joan. Joey is pleased to see her. Joan says his father was a good man. Joey says she was a good doctor. He offers to buy her coffee to catch up.
Sherlock meets Gregson and Bell at the crime scene. The dead man in the apartment is Felix Soto who was a math tutor. The mugger was named Bennie. He's in surgery and it doesn't look good. One room in Soto's house is covered with math equations that are only visible in black light.
Over coffee, Joey tells Joan thay he quit college because he was having a tough junior year. But he's about to buy a bar. He hits her up for an investment in the bar. She asks how much he needs.
Back at home Sherlock has recreated the black light and equations on his walls and hired a shirtless mathematician named Harland to look at them. Harland is a little wacky; he's a math genius.
Joan asks Sherlock for an advance on her pay.
Harland explains what the equations are about; it's a famous problem called P vs NP that involves problems being fixed by computers. Some experts say that P vs NP can't be solved. Harland thinks the mathematicians involved here, Soto and his shooter, were close to solving the problem. He also thinks they were probably competing for a prestigious and lucrative prize.
Sherlock and Watson meet with Tanya, a woman who interviewed people in the P vs NP community which was highly competitive. She gives them names of people she talked to for her paper and one matches the handwriting to the likely shooter, a guy named Cyril. The only problem is when Watson and Sherlock are about to look him up, Bell calls and tells them Cyril is dead, shot.
They examine the bodies side by side in the morgue and determine they were killed by the same shooter and figure it was a competitor. Some hairs on the second body make it seem as if the shooter has a dog.
Watson asks Sherlock again about the money. He asks for more details, knowing she went to visit her former patient's grave and it's his son asking her for money. She stonewalls. She notes they never really talked about her patient who died on her table. She explains the surgery and how it went wrong and how it was her fault and how he was a nice man, a dockworker. She got to know his family a few weeks before the surgery. The wife sued her and said terrible things to her which was difficult of course. The son, who was 17 at the time, wrote her a letter saying that he forgave her. It meant a lot to her at the time. This isn't the first time he's asked her for money. She helped him buy a car.
Bell calls from Cyril's house and says there's stuff that they need to see including a bug that was listening in on the dead man and his research. Sherlock notes a security camera across the street that has been hijacked to look into the guy's house.
They go to trace it. It leads to a man at an encryption service. He admits that they were monitoring Cyril regarding P vs. NP. They weren't after the prize money but the fact that a solution would have massive real world applications and would render modern encryption obsolete. He says he wanted Cyril to solve it and offered to fund his work so he could write code that could defeat the solution so his company could stay in business and also, of course, make lots and lots of money. But Cyril didn't want money, he wanted acclaim. The encryption guy also infiltrated Felix Soto's house and says they discovered they weren't close to solving it. Which is of course a lie. He went to Tanya, the same woman who told Holmes and Watson they were very close.
They bring Tanya in and she admits to lying to the encryption guy and obfuscating the truth when she spoke to Holmes and Watson. She claims they still had years of work ahead of them and figured if she lied to the encryption guy it would protect them. It turns out that she owns a dog and a gun, so she might be the shooter. They theorize she wanted to finish the work and sell the solution to the encryption guy herself. She gives an alibi of a date at a restaurant. They get security footage from the restaurant and her alibi holds up.
Watson and Holmes look at the pics and CVs (resumes) of other people trying to solve P vs. NP who might have also been watching Felix and Cyril. Sherlock also agrees to give Joan the money against his better judgement. He gives her $20,000 instead of 5; he wants her to give it all to Joey so he never asks her for money again. He calls it his gift to her to put her mistake in the past. They get back to the case and wonder if someone is trying to frame Tanya.
Sherlock talks to her at school the next day and asks if she might have enemies. She thinks it might be her ex-boyfriend Jason. The cops bring Jason in. He claims to know nothing but they pull up evidence that he did know them and that he ordered 9 mm ammo.
They learn the mugger that was shot in the alley during Felix's murder is now awake. He says a woman shot him. It was Tanya.
Sherlock again asks Watson to pay off Joey. She demurs.
They realize Tanya manipulated the security footage date/time stamp, because she had solved P vs. NP but was trying to buy time with a computer programmer to write code to make it work. That computer programmer was her date and ratted her out.
Wason meets with Joey. She tells him she's investing in him, not the bar, and giving him money to go back to school. He doesn't want to do it. She says it's the only help she's comfortable offering. He gets angry and blames her for his dad's death. He apologizes. She says if he opens a bar she'll come, she'll talk to him anytime, but she's only going to pay for his education.
Watson comes home and finds Sherlock studying the P vs NP equations one last time, because the NSA wants the equations and Tanya in custody for national security reasons.
Watson tells him that Joey is considering her offer. He is proud of her. She thanks him for offering the money. He says he wants to go with her the next time she visits the cemetery since that man obviously changed the course of her life and brought her to him and he'd like to pay his respects. She says she'd like that.
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