"The Final Problem" is the third episode of the fourth series of the British television series Sherlock, and the thirteenth episode overall. The episode first aired on BBC One, PBS, Channel One and 1+1 on 15 January 2017. To date, no further episodes of Sherlock have been produced, making "The Final Problem" the presumptive series finale.
Plot
A young girl wakes up on an aeroplane and finds everybody asleep. Panic stricken, she answers a ringing mobile phone, only to hear the voice of Jim Moriarty announce, "Welcome to the final problem".
After Sherlock stages a break-in at Mycroft's home and tricks Mycroft into revealing Eurus' existence, Mycroft reveals that, like himself and Sherlock, Eurus was also born with advanced intellectual abilities, even describing her as an era-defining genius on a par with Isaac Newton. However, she also has a total lack of normal sensation and emotion.
After she kidnapped and killed Sherlock's dog, Redbeard, and set their house on fire, Sherlock's parents sent Eurus to a mental institution. But Mycroft reveals that, behind their parents' backs, he had Eurus transferred to Sherrinford, a government "black site" and maximum-security "prison within a prison" in the North Sea. The whole experience had been so deeply traumatising to Sherlock that his memories of her had been “rewritten”.
Their conversation is interrupted when a quadcopter carrying a motion-activated grenade flies into Sherlock's flat. John, Sherlock, and Mycroft flee as the bomb detonates, blasting Holmes and Watson out of the windows. Later, John and Sherlock hijack a fishing trawler to travel to Sherrinford, carrying out a diversionary plan so that Sherlock can reach Eurus' cell with Mycroft disguising himself as a fisherman.
Mycroft and John corner the prison governor, discovering that he has explicitly disobeyed Mycroft's protocol and has allowed Eurus to interact with prison staff. Using her skill to "reprogram" everyone she speaks with, Eurus has effectively ruled the prison. Meanwhile, Sherlock talks to Eurus; but after toying with him, particularly about Redbeard's death, she attacks him and knocks him unconscious. The guards lock Sherlock, John, Mycroft, and the governor together in Eurus' old cell. It is then revealed that five years prior, Mycroft granted Eurus an unsupervised five-minute interview with Moriarty as a Christmas present, in exchange for her detecting national security threats to Britain. During that time, Moriarty agreed to record video messages for her.
John wakes up in Eurus’ cell with Sherlock, Mycroft, and the governor, and the young girl's distress call comes through the speakers. Sherlock tries to talk to the girl, but Eurus stops the call. After forcing the governor to kill himself, Eurus mentally torments Sherlock, Mycroft, and Watson, forcing them into cruel and deadly games to save their lives while videos of Moriarty taunt them. Although Eurus forces Sherlock to continue with the prospect of saving the girl on the aeroplane, he eventually stops the games by threatening to shoot himself when she orders him to murder either John or Mycroft. Furious, Eurus has the three of them shot with tranquilliser darts.
Sherlock wakes up near the burnt-out wreckage of his family's old house. He speaks to the girl in the aeroplane to guide her in landing safely. John wakes up chained at the bottom of a well. As Eurus raises the water level in the well, John finds a child's skull there. Sherlock realizes that what he thought was his dog Redbeard was his childhood friend, Victor Trevor. Redbeard was Trevor's name when they played as pirates, and Eurus threw him into the well and left him for dead because she felt left out of Sherlock's game.
Sherlock then figures out where Eurus is hiding by deciphering the real meaning of the song that Eurus had taunted him with when Victor went missing, which reveals that she wants him to find her. It is also revealed that the little girl on the aeroplane has been Eurus all along, and the game was a cry for help, for whenever she closes her eyes she's on the plane and all alone. After helping her and rescuing John, Sherlock sends her back to Sherrinford.
Mycroft explains to his and Sherlock's parents, who are furious that he told them Eurus was dead, that she refuses to speak to people anymore. Sherlock visits Eurus, and they play the violin together instead of talking. While helping Sherlock repair his destroyed flat, John finds another video sent by Mary before she died. Sherlock and John watch as Mary explains posthumously that, whatever kind of person they may be, they will always be her Baker Street Boys – "Sherlock Holmes and Doctor Watson", the series ending with the flat restored and the two resuming to their lives of solving mysteries.