Hercule Poirot, the celebrated Belgian detective, is interrupted at breakfast by a disheveled young woman who tells his manservant, George, that she wants to consult Poirot about a murder she might have committed. After staring at him, the girl abruptly blurts out that he is "too old" and flees. Poirot is wounded but far more alarmed: A girl seeking help about a possible murder has left without receiving any.
Poirot confides in his friend, the detective novelist Ariadne Oliver, who realizes she may have sent the girl to Poirot by praising him at a party. Through inquiries, Mrs. Oliver identifies the girl as Norma Restarick, stepdaughter of Mary Restarick and daughter of Andrew Restarick, a wealthy businessman recently returned to England after years abroad. Norma lives at Borodene Mansions in London, sharing a flat as "the third girl" with Claudia Reece-Holland, the capable daughter of a Member of Parliament, and Frances Cary, a languid, black-haired young woman who works for an art gallery. Local gossip holds that Mary has suffered a mysterious gastric illness, and neighbors whisper about poisoning. Norma's boyfriend, David Baker, a strikingly handsome young man Mrs. Oliver dubs "the Peacock," has been forbidden from the house by her father. Poirot engages Mr. Goby, a discreet private investigator, and contacts Scotland Yard to search for unsolved deaths.
Mrs. Oliver visits the flat and discovers Norma has not returned from a weekend at home. Poirot travels to Crosshedges, a house in the village of Long Basing belonging to Sir Roderick Horsefield, a distinguished but elderly and nearly blind uncle by marriage with whom the Restaricks are staying. Posing as an old acquaintance of Sir Roderick's, Poirot meets Mary in the garden, a handsome golden-haired woman whose hair he suspects is a wig. He encounters David, who has entered uninvited looking for Norma, and meets Sonia, Sir Roderick's young foreign secretary-companion. Unobserved, Poirot finds fresh mud in Norma's bedroom suggesting David searched the room.
Mr. Goby's reports reveal troubling details: A porter once found Norma dazed in the courtyard holding a revolver, with blood on the ground, and Claudia quietly suppressed the incident. Andrew Restarick's history includes abandoning his first wife, Grace, and their young daughter years earlier to run off to South Africa with a woman named Louise Birell. Separately, Frances confides to Claudia that she found a flick-knife with dried blood hidden in Norma's drawer, and that the knife later vanished.
Mrs. Oliver stumbles upon Norma and David in a café. Disguised, she overhears Norma confess that she hates her stepmother, found weed killer in her drawer, and experiences terrifying blank periods during which she cannot remember her actions. David proposes marriage as a way to free her from her family's control. Poirot arrives after Mrs. Oliver leaves to shadow David and sits with Norma, but she is terrified of doctors and refuses help. Mrs. Oliver trails David to an artist's studio in Chelsea where Frances is posing for a painter; after leaving and getting lost near the river, Mrs. Oliver is struck on the head and hospitalized.
Meanwhile, Norma nearly steps in front of a speeding car in what appears to be a suicide attempt. John Stillingfleet, a blunt, red-haired psychiatrist, pulls her to safety and persuades her to stay at a quiet rest home called Kenway Court. That evening, Stillingfleet telephones Poirot: Someone has been administering a cocktail of drugs including LSD and hemp to Norma without her knowledge, producing hallucinations, memory blackouts, and distorted perceptions of reality.
Poirot visits Andrew Restarick at his office after receiving a forged letter requesting a meeting. With Norma missing and the police out of the question, Restarick hires Poirot to find her. He admits Mary's illness was traced to deliberate poisoning, with a toxic substance found among Norma's belongings. Poirot notices a large cheque made out to David, suggesting a payoff, and studies the Lansberger portrait of Restarick behind the desk, one of a pair originally displayed at Crosshedges. Sir Roderick also visits Poirot to report missing wartime papers, and Goby reveals that Sonia was observed at Kew Gardens in what appeared to be a covert exchange with a junior attaché from a foreign embassy.
Mrs. Oliver, recovering in hospital, mentions a forgotten detail: A woman named Mrs. Charpentier recently fell from a seventh-floor window at Borodene Mansions, roughly a week before Norma came to Poirot. The death was ruled suicide. Poirot investigates and notices that the flat numbers are attached with spikes and come loose easily. Chief Inspector Neele of Scotland Yard reveals that Louise Charpentier was once the mistress of Claudia's father, and Poirot realizes she was almost certainly Louise Birell, the woman who ran off with Andrew Restarick years earlier and the one person who could recognize whether the man now claiming to be Restarick was genuine.
Norma's former headmistress, Miss Battersby, declares her "an emotional but normal girl" (214). Sitting in his armchair, Poirot assembles the pieces: the wig, the portrait, the "Third Girl" who was never quite present, the loose door numbers, and the early-morning timing of Louise's death. The pattern clicks. Frances Cary and Mary Restarick are the same woman, maintaining two identities to provide a perpetual alibi. Before Poirot can act, Stillingfleet telephones: Norma has left Kenway Court after reading a coded message in the Personal columns that lures her to the flat.
At Borodene Mansions, Frances returns to find David dead on the floor, stabbed. A neighbor, Miss Jacobs, finds Norma holding a kitchen knife, calmly saying she killed him. The police arrive, along with Restarick, Claudia, Poirot, Mrs. Oliver, and Stillingfleet. During the commotion, Poirot briefly disappears and slips a black folder into Mrs. Oliver's shopping bag.
At the inquiry, Stillingfleet declares Norma completely sane: Someone deliberately administered drugs to make her believe she was losing her mind. Poirot asks Mrs. Oliver to produce the folder's contents: a wig of bouffant golden hair retrieved from Frances's overnight bag. Placed on Frances, it transforms her into Mary Restarick. Neele addresses Restarick as "Robert Orwell," revealing him as an impostor. Poirot explains that the real Andrew Restarick died abroad, and Orwell, a fellow adventurer who had traveled with the real Restarick in Africa, assumed his identity to claim the enormous Restarick fortune. Frances was his accomplice, playing "Mary" in the country and "Frances" in London. David was killed because, after forging the portrait that served as false proof of Orwell's identity, he began blackmailing the pair. Louise was murdered because she would have recognized the impostor; Frances pushed her from the window at five in the morning, having switched door numbers so the drugged Norma would believe she had been at Louise's door.
Poirot comforts Norma, telling her she is neither mad nor a murderer but the intended victim of two ruthless conspirators. Stillingfleet proposes marriage with characteristic bluntness, telling Norma he is leaving for Australia and she should join him. She smiles and agrees. Norma returns to Poirot, apologizes for calling him too old, and kisses him. Mrs. Oliver asks privately whether Poirot deliberately paired Norma with Stillingfleet; he admits he did, and Mrs. Oliver departs: "You do think of things, don't you" (275).