Elspeth McGillicuddy has come down from Scotland to visit her old friend Jane Marple. On the way she sees a woman strangled in a passing train. Only Miss Marple believes her story as there is no evidence of wrongdoing. The first task is to ascertain where the body could have been hidden. Comparison of the facts of the murder with the train timetable and the local geography lead to the grounds of Rutherford Hall as the only possible location: it is shielded from the surrounding community, the railway abuts the grounds, and so on. Miss Marple calls upon an acquaintance, Lucy Eyelesbarrow, who is a professional housekeeper renowned for her efficiency and organisational skills. Lucy agrees to take a position at Rutherford Hall and the hunt is on.
Rutherford Hall was built by Josiah Crackenthorpe, purveyor of tea biscuits. His son Luther, now a semi-invalid widower, had displayed spendthrift qualities in his youth. To preserve the family fortune, Josiah left his considerable fortune in trust, the income from it to be paid to Luther for life, and after Luther's death the capital to be divided equally among Luther's children. Although Luther Crackenthorpe is the present occupant of Rutherford Hall, he cannot sell the house as per the terms of his father's will. The house will be inherited by Luther Crackenthorpe's eldest surviving son or his issue.
The eldest of Luther Crackenthorpe's children, Edmund, died during World War II. His youngest, daughter Edith, died four years before. The remaining heirs to the estate are Cedric, a painter and lover of women who lives on Ibiza; Harold, a cold and stuffy banker; Alfred, the black sheep of the family and a man known to engage in shady business dealings; Emma Crackenthorpe, a spinster who lives at home and takes care of Luther; and Alexander, son of Edith. The characters are completed by Bryan Eastley, Alexander's father and Dr. Quimper,who looks after Luther's health and is quietly romantically involved with Emma.
Lucy uses golf practice as an excuse to search the grounds. She eventually finds the woman's body hidden in a sarcophagus in the old stables amongst Luther's collection of dubious antiques. But who is she?
The police identify the victim's clothing as being of French manufacture. Emma tells the police that she has received a letter claiming to be from Martine, a French girl whom her brother had wanted to marry. He had written about Martine and their impending marriage just days before his death in the war. The letter purporting to be from Martine claims that she was pregnant when Edmund died and that she now wishes their son to have all of the advantages to which his parentage should entitle him. The police conclude that the body in the sarcophagus is Martine but this is struck down by Lady Stoddart-West, mother of James Stoddart-West, friend of Alexander's. Lady Stoddart-West reveals she is Martine. Although she and Edmund had intended to marry, Edmund died before they could do so.
The plot thickens when the whole family takes ill suddenly and Alfred dies. Later, the curry made by Lucy on the fateful day is found to contain arsenic. Some days later, Harold, after returning home to London, receives a delivery of some tablets that appear to be the same as the sleeping pills prescribed to him by Dr Quimper, but who had told him he need not take them anymore . They prove to be poisoned and Harold dies. One by one the heirs to Josiah's fortune are vanishing.
Lucy arranges another tea time visit to Rutherford Hall for Miss Marple and Mrs McGillicuddy is invited down from Scotland. She is instructed by Miss Marple to ask to use the lavatory as soon as they arrive but is not told why.
Miss Marple is eating a fish sandwich when she begins to choke. It seems she has a fish bone stuck in her throat. Dr Quimper moves to assist her. Mrs McGillicuddy enters the room at that moment, sees the doctor's hands at Miss Marple's throat, and cries out 'but that's him - that's the man on the train!' Miss Marple had correctly concluded her friend would recognise the real murderer if she saw him again in a similar pose.
It transpires that the murdered woman had been married to Dr Quimper many years earlier. He murdered her so he would be free to marry Emma and inherit Josiah's fortune once he got rid of all the other heirs.