It's a math-themed evening of Elementary tonight -- or maths, as the British say. Sherlock is investigating the murder of a brilliant mathematician, while Watson is grappling with a tricky guilt to generosity ratio, as the son of a former patient keeps subtly exploiting her for money.
The episode begins with a common thief witnessing a murder while mugging someone, bellowing to the rafters that he doesn't want any trouble, and getting shot himself. Though the initial homicide was successful, the thief survived in critical condition--he's comatose when Sherlock arrives to investigate. Sherlock finds traces of an invisible ink only visible under ultraviolet light, revealing a complex smorgasbord of mathy-looking diagrams on the victim's wall.
Watson, meanwhile, is busy catching up with an old friend, the son of a former patient. The young man invites her out for coffee, where he offers her an opportunity to invest in a bar he's trying to operate a.k.a. asking for a loan. Watson isn't entirely sure about the idea, but she agrees to look into it.
With the help of an outside consultant, Sherlock discovers that the equations on the wall are a nearly-completed mathematical proof that many had thought impossible -- if completed, it would be worth at least a million dollars. The detectives speculate that the deceased mathematician might have been betrayed by a colleague. When they discover who his partner was, however, they learn that he was killed as well.
As the investigation continues, Watson reveals to Sherlock that her potential investor's father had died under her care. Though his death was the result of a mistake that Watson made, the son sent her a letter absolving her of any wrongdoing. It meant a lot to her, but he's subtly manipulated her since then.
Sherlock and Watson discover that the mathematician was being spied on by a major security company, as the equation would have major implications for cryptography. The Company's C.E.O. reveals that their interest had waned, however, when an outside consultant had declared their work bogus; this was a lie making the consultant their new chief suspect. She had a handgun, she had a pet dog whose fur matched hair found on the victim, but she also has a video-taped alibi placing her at a bar during the murder.
Considering how thoroughly the evidence points to her -- and how impossible it is that she might have done it -- Sherlock begins to suspect that she might have been framed. It turns out that she had gone through a nasty split with her old boyfriend, and he had sent her a number of threatening e-mails. With the help of a warrant, they're able to trace an interest in the equation and an order for ammunition to his e-mail account, though he quickly denies that he's ever sent any messages to that effect.
At the same time, however, the thief shot in the opening scene wakes up and I.D.s a female shooter, bringing the consultant back as the chief suspect. Reevaluating the security footage that had absolved her, Sherlock and Watson discover that bar patrons near her had been paying happy hour prices, inconsistent with the digital time stamp. They deduce that she had solved the cryptographically valuable equation, and had used it to hack both the security camera and her ex's e-mail.
On a personal note, Sherlock helps absolve Watson's guilt by fronting her a $22,000 advance to buy out the young entrepreneur's continued manipulation. She shows her good will by offering him the money for his education -- it's what his late father would have wanted -- but refusing to continue indulging him with no strings attached.
Though tonight's central mystery wasn't quite a full force mental triumph -- simply changing a digital time stamp seems a lot more intelligent when you surround it with equations and cryptography -- it was interesting to see Watson grappling with the risk and loss of her medical practice. I doubt that most doctors feel obligated to put their deceased patient's families through college, but it's still a reminder that she went through some pretty harrowing life experiences before ever meeting Sherlock. That he would support her unconditionally even while knowing why she's wrong is a testament to how far their friendship has come.
(Image courtesy of CBS)