The novel opens as the family and the victim are introduced through the perspective of Sarah King and Dr. Gerard, who discuss the behavior of the family. Mrs. Boynton is sadistic and domineering, which she may have inculcated from her original profession: prison warden. Sarah is attracted to Raymond Boynton, while Jefferson Cope admits to wanting to take Nadine Boynton away from her husband, Lennox Boynton, and the influence of her mother-in-law. Having been thwarted in her desire to free the young Boyntons, Sarah confronts Mrs. Boynton whose apparent reply is a strange threat: "I’ve never forgotten anything – not an action, not a name, not a face." When the party reaches Petra, Mrs. Boynton uncharacteristically sends her family away from her for a period. Later, she is found dead with a needle puncture in her wrist.
Poirot claims that he can solve the mystery within twenty-four hours simply by interviewing the suspects. During these interviews he establishes a timeline that seems impossible: Sarah King places the time of death considerably before the times at which various of the family members claim last to have seen the victim alive. Attention is focused on a hypodermic syringe that has seemingly been stolen from Dr. Gerard’s tent and later replaced. The poison administered to the victim is believed to be digitoxin, something that she already took medicinally.
Poirot then calls for a meeting and explains how each member of the family has, in turn, discovered Mrs. Boynton to be dead and, suspecting another family member, failed to report the fact. None of the family would have needed to murder the victim with a hypodermic, since an overdose could much more effectively have been administered in her medicine. This places the suspicion on one of the outsiders.
The murderer is revealed to be Lady Westholme who, previous to her marriage, had been incarcerated in the prison in which the victim was once a warden. It was to Lady Westholme, and not to Sarah, that Mrs. Boynton had addressed that peculiar threat; the temptation to acquire a new subject to torture had been too great for her to resist. Disguised as an Arab servant she had committed the murder and then relied upon the suggestibility of Miss Pierce to lay two pieces of misdirection that had concealed her role in the murder. Lady Westholme, eavesdropping in an adjoining room, overhears that her criminal history is about to be revealed to the world and commits suicide. The family, free at last, take up happier lives: Sarah marries Raymond; Carol marries Jefferson; and Ginevra takes up a successful career as a stage actress – she also marries Dr. Gerard.