Lieutenant Colin Race, a reluctant spy for the Royal Navy, tracks down the celebrated sleuth, Hercule Poirot, to relate a tale of murder and espionage. Race first became embroiled in the situation when a beautiful stranger ran shrieking into his arms in a neighborhood street, horrified at having discovered a corpse in the house where she'd been requested for a job. The woman, a young typist named Sheila Webb, is now the main suspect in an intricate puzzle whose only apparent clue is the inexplicable placement of four clocks around the crime scene, none of which belong to the house's owner, all frozen at exactly 4:13. Without an alibi and unable to explain away the clocks, Sheila is dependent on Race — and the meticulous Belgian brain of Poirot — to exonerate her and find the murderer.
Yet Race, under pressure from his superiors at the Royal Navy to solve a parallel — or perhaps connected — investigation of a dangerous German mole, has lost his objectivity, too much the gallant champion of a woman whom he considers innocent. It is up to Poirot to take on the eccentrics of the crime scene's neighborhood — the extreme cat fancier Mrs. Hemmings, the milquetoast Blands, the arrogant Mr. Mabbutt, and the academic Waterhouse siblings. But the case seems only to further confound, and no one can even establish the identity of the victim...until a second victim is discovered. The clock is ticking down as Poirot strives to solve the case before the Germans can exploit their bloody intelligence and before the murderer strikes again.
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Warning: Contains significant plot spoilers
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Deep in the underground tunnels of Dover Castle during the buildup to war with Germany, Annabel Larkin secretly tucks stolen plans into an umbrella handle's hidden compartment. Undercover British agent Fiona Hanbury watches from the shadows, then calls her fellow agent and fiancé Lieutenant Colin Race for backup. But Race, gambling, dismisses the call. The German mole leaves, Fiona pursues her, and in a tussle both women are hit by a car and end up dead.
Burdened by guilt, Race follows his sole clue to the quiet neighborhood of Wilbraham Crescent. There, he encounters a screaming woman fleeing a home, begging for help. The beautiful young typist, Sheila Webb, found a corpse at her feet in the eerily empty house where she'd been sent for a secretarial job. Her proximity makes her the prime suspect and it's up to the chivalrous Race to exonerate her.
In London, Race tracks down his father's old friend Hercule Poirot and requests the Belgian sleuth's help. Race relates the details of the case: Sheila had been requested specifically for the job, yet both she and the house's elderly, blind owner, Miss Pebmarsh, claim no prior knowledge of one another; four clocks were placed around the crime scene, all frozen at precisely the same moment in time, 4:13; and one of the clocks, upon which is written "Rosemary," is missing. Miss Pebmarsh has never before seen the victim, or the puzzling four clocks. Nor did she call for a secretary.
In Dover, Poirot and Race join forces with Inspector Hardcastle and are briefed on case by Vice Admiral Hamling of the British Navy, who believes that the German spy and the dead man are connected. Confiding in Poirot, he underscores how disastrous it would be if the stolen plans, which reveal England's minefields in the Channel, fell into German hands.
Learning that the victim's business card is a fake, Poirot and Race now must add the victim's identity to their list of unanswered questions. They interview Miss Pebmarsh's neighbors: the strange cat lady Mrs. Hemmings, the academic siblings the Waterhouses, and the innocuous Blands, who fairly recently came into an inheritance. Next, they speak with Mr. Mabbutt, who frequently travels to the continent on behalf of Armstrong Ordinance, working with the French army. At the photo studio where she works, Poirot interviews Miss Pebmarsh, who was blinded by a shell blast in WWI. He then endures the endlessly chatty secretary Nora at the Cavendish Secretarial Bureau, Sheila's workplace, to meet with Miss Martindale, the agency proprietress. He learns that Miss Martindale was once the private secretary to the great mystery writer Garry Gregson.
During the investigation, Sheila, alone in the world, has become close with the grieving Race, and an attraction has blossomed. Now about to depart for the inquest, Sheila receives an anonymous postcard inscribed "Remember 4:13." The testimonies of Miss Martindale and Miss Pebmarsh reveal nothing new; nevertheless, Nora is agitated. As everyone leaves after the inquest, she attempts to tell Hardcastle that "...what she said couldn't possibly be true" but is brushed off by one of the inspector's men.
Evidence at the inquest has piled up against Sheila, and Val Bland stops by to comfort her. But Mr. Bland informs Poirot that he saw Sheila and the victim together at the Castle Hotel the day before the murder. Professing Sheila's innocence, Race is accused by Hardcastle of having lost his objectivity. But Sheila is at the Castle Hotel, arguing with a man and trying to leave while he begs her to come back inside the room. The room number, Poirot later learns, is 413.
Meanwhile, Nora anxiously calls police headquarters to inform Hardcastle about the lie she'd heard at the inquest. She never gets to speak to him: she is found murdered in the phone box, strangled by the phone cord.
Poirot learns from the anti-war Miss Pebmarsh that her sons had been killed in WWI. Miss Martindale reveals that Sheila kept a regular appointment at the Castle Hotel with a certain Professor Purdy, implying that Sheila was performing questionable "work" there. Race, reuniting with Sheila, is horrified to find one of the clocks and the bloody murder weapon in her bag. But at police headquarters, Poirot reveals that the clock was Sheila's, disappeared from the shop she had brought it for repairs. She had taken it from the crime scene, panicked that she was being framed for the crime. The knife, he asserts, she had never seen before; otherwise she would have disposed of it. He focuses on Nora, whom Sheila had witnessed complaining about breaking her heel in a rain grate en route to lunch the day of the murder. Declaring Sheila's innocence, Poirot asks a stunned Hardcastle to release her from custody.
Meanwhile, an apparent breakthrough in the case occurs in the form of Merlina Rival, the widow of the murdered man. She identifies him, noting an old scar and qualifying him as no-good. But tipsy at the pub, she is soon caught in a lie by Hardcastle. She makes a call, demanding more money and agreeing to meet her interlocutor. Hardcastle follows, but loses her when she turns a corner. They soon find her murdered.
Mabbutt's daughters proudly show Poirot the umbrella they unearthed in their own yard. He urges Race to prevent Mabbutt from leaving the country and orchestrates the arrests of the spies Mabbutt and Miss Pebmarsh for the crime of high treason. Poirot reveals that Annabel Larkin would steal the information, Miss Pebmarsh would photograph it, and Mabbutt would deliver it to the enemy, all to weaken England in order to prevent the country from entering war. Miss Pebmarsh above all wished to preserve the peace and prevent deaths like her sons', and Mabbutt to join forces with Germany against the spread of Communism.
Poirot then assembles the remaining suspects together at the Cavendish agency where he conjures up Nora, who had returned early from lunch because she broke her heel. She was present to observe that the phone never rang, and therefore she had to be silenced. The murder victim, he reveals, was the uncle of the real Mrs. Bland, whose identity was stolen upon her death so that the couple could collect her inheritance. Because the uncle had come from Canada, they could easily kill him to protect their secret and frame a young woman whom they saw as disposable. Their murder plot was cribbed from a story by the mystery writer Gary Greggson; Miss Martindale, the imposter's sister, conceived of the plan and provided a dupe, Sheila Webb. Sheila's own clock and the veiled reference to room 413 were present to frighten and threaten the girl into silence, but that wasn't enough, so Val Bland slipped the murder weapon into Sheila's bag after the inquest.
As the murderers are taken away, the broken Sheila takes her leave. It's Poirot who urges Race to follow his heart and go after her.