'Elementary' Clue-cap: Sherlock and Watson Run in Circles, then 'Solve for X'

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Thursday's "Elementary" episode, "Solve for X," had Sherlock (Jonny Lee Miller) and Watson (Lucy Liu) following clues in search of a killer math whiz. Where did those clues lead? Well, in circles, until a final clue broke the cycle. 

A man named Felix Soto is murdered, and a mugger — who happened to be in the wrong place at the wrong time — is left for dead, unlikely to ever wake up.

Sherlock and Watson encounter their first set of clues:

Clue 1: An empty room in Felix's home, devoid of wall decorations or furniture.

• The walls smell like spent matches.
• The light switch does not work.

The clues lead Sherlock to believe that the walls are covered with invisible ink. A hidden black light lamp in the other room illuminates a multitude of math problems.    

Sherlock discovers the equation:

A friend of Sherlock's reveals that the scribblings are the near solution to the P versus NP problem, worth a million dollars to whoever  can solve it. It could also theoretically be used, given enough time, to write code that can hack any server on the planet.

Sherlock's buddy figures out what the equation is:

Clue 2: The math on the wall was written by two men, based on two distinct styles of handwriting.

Clue 3: A photo of a man in a book — in which P versus NP expert Tanya Barrett (Lynn Collins) profiled Felix and other P versus NP fanatics — includes a chalkboard with handwriting matching that of the second math genius who wrote on the wall. 

Watch Clue 3 go down:

Cyril Nauer (the man from the book) is found dead, as well, along with an additional set of clues:

Clue 4: Dog hairs are found under the arms of Cyril's coat.

Clue 5: A bug is found in a fixture and traced to an encryption company.

Linus Roe from Roe Encryption Technologies, the company spying on Felix and Cyril, did not want the two men dead. Tanya, who profiled the two men, lied to Roe, saying that the men were nowhere close to solving P versus NP.

The dog hairs and a 9 mm handgun reported missing lead back to Tanya. In addition, the man left for dead wakes and claims that Tanya shot him. 

The injured man IDs Barrett:

But Tanya has an alibi: A surveillance camera at a diner shows her having dinner at the time of the murder.

The team is duped by the surveillance footage:

Sherlock and Watson are stuck going in circles until they find one final clue that breaks the loopy conundrum:

Clue 6: A man on the diner surveillance tape pays happy hour prices for four beers, but the digital timestamp on the tape indicates that it is no longer happy hour.

This leads Sherlock to realize that Tanya had solved P versus NP on her own and used it to hack the surveillance camera at the diner. She then killed Felix and Cyril before they could reach the solution, so she would have time to use it to steal millions of dollars.

Mystery solved! 

Sherlock and Watson expose Barrett:

Meanwhile, Watson was trying to solve her own mystery of whether to help Joey Castoro (Jeremy Jordan), a young man whose father, Gerald, had died on Watson's operating table years ago. Sherlock points out the one clue that suggests Joey may not be sincere: He is asking for $5,000, and this is the second time he has asked for a large sum of money.

Watson's solution to the mystery is to give Joey a $20,000 scholarship — money given to her by Sherlock to "buy off" Joey so he won't bother her anymore. When Joey's words turn from kind to hurtful, and he doesn't sound as if he will accept the offer, her own mystery is solved. 

Sherlock addresses the Joey situation:

"Elementary" airs Thursdays at 10 p.m. on CBS.

Check out episode photos from "Elementary":