Mr Shaitana, a flamboyant collector, meets Hercule Poirot by chance at an art exhibition and brags about his personal crime-related collection. Scoffing at the idea of collecting mere artefacts, Shaitana explains that he collects only the best exhibits: criminals who have evaded justice. He invites Poirot to a dinner party to meet them.
Poirot's fellow guests include three other crime professionals: secret serviceman Colonel Race, mystery writer Mrs Ariadne Oliver, and Superintendent Battle of Scotland Yard; along with four people Shaitana believes to be murderers: Dr Roberts, Mrs Lorrimer, Anne Meredith, and Major Despard. Shaitana taunts his suspects with comments that each understands as applying only to them.
The guests retire to play bridge, the professionals playing in one room while the others play in a second room where Shaitana relaxes by the fire. As the party breaks up, Shaitana is found to be dead – stabbed in the chest with a stiletto from his own collection. None of the suspects can be ruled out, as all had moved around during the evening. Leading the police investigation, Superintendent Battle agrees to put his "cards on the table" and to allow the other professionals to make their own enquiries. Poirot uses the bridge scores to deduce the playing style and personality of each suspect.
The investigators look into the suspects' histories. The husband of one of Dr Roberts' patients died of anthrax shortly after accusing the doctor of improper conduct. A botanist that Despard had been guiding through the Amazon was rumoured to have been shot. Anne's housemate Rhoda Dawes tells Mrs Oliver that an elderly woman for whom Anne was acting as companion died after mistaking hat paint for syrup of figs. Mrs Lorrimer's husband had died twenty years earlier, though little is known about that.
Mrs Lorrimer tells Poirot that she has just been diagnosed with a terminal illness, and that she wishes to confess to killing both her husband and Shaitana. Poirot refuses to believe her psychologically capable of spontaneous murder, and thinks that she is protecting Anne. Mrs Lorrimer reluctantly discloses that she had actually seen Anne commit the crime, but feels sympathy for a young girl just starting out in life. The next day, Despard, Roberts, and Anne all receive a suicide note and confession to the crime, supposedly written by Mrs Lorrimer. Battle informs Poirot by telephone that Dr Roberts rushed to the house only to find the woman had died of an overdose.
Poirot is again suspicious, as he knows that Anne had visited Mrs Lorrimer the previous night. He discovers that, due to the established time of death, Mrs Lorrimer could not possibly have sent the letters. Realising that Rhoda's life is in danger, since she knows about the syrup of figs incident, Poirot, Battle, and Despard race to Rhoda's cottage, arriving to find the two girls in a boat out on the river. Anne attempts to push Rhoda overboard, but Anne herself falls in and drowns, while Rhoda is rescued by Despard.
Poirot explains his findings. Despard had indeed shot and killed the botanist, but entirely by accident. Anne had poisoned her employer by switching the two bottles, to conceal her petty thieving. Although Mrs Lorrimer thought she had seen Anne kill Shaitana, Anne had in fact just leaned forward to touch him and confirm he was already dead.
Poirot notes that only one suspect was psychologically capable of carrying out such a risky, unplanned stabbing – namely Dr Roberts. Fearing Shaitana had proof of the anthrax murder, Roberts quickly took his chance. He covered his tracks by forging Mrs Lorrimer's letters and killing her with an injection when he visited her house. Although Roberts initially protests, he is forced to confess when Poirot reveals a surprise eye-witness to the killing, a window-cleaner. After Roberts is led away, Rhoda notes what amazing luck it was that the window cleaner had been there at the exact moment of the fatal injection. Poirot replies that it had not been luck at all, and introduces them to a hired actor whose presence had prompted Roberts' confession.
With the murder solved, Despard courts Rhoda.
