Miriam Granger is vulgar, spoiled and known to sleep around a lot. This is why her husband, amateur tennis player Guy Haines, wants to divorce her; at least, that is part of the reason. Another motivation is his love for Anne Morton, who seems to be everything that Miriam is not. Elegant and well brought up, Anne's father is a Senator and she has enjoyed somewhat of a charmed life.
Guy meets Bruno Anthony on a train; Anthony is a dangerous man, a psychopath, and the two discuss ideas for the perfect murder. What if two complete strangers were to meet on a train, and each kills the person that the other wants dead? That way, there is no motive and no connection that will tie them to the victim. Anthony will kill Miriam, and Guy will kill Anthony's father, whom he hates. Unlike Anthony, Guy is not a psychopath, and so feigns amusement at the idea without actually agreeing to it, but Anthony, who cannot read social cues properly, mistakes this amusement for agreement to participate in the plan. In his haste to leave, Guy leaves behind a monogrammed cigarette lighter with the initials A to G engraved on it, above crossed tennis rackets. Anthony sees it, and keeps it.
Miriam, who is pregnant with another man's child, tells Gary that she no longer wants a divorce. She intends to tell everyone that the baby is his so that he will not be able to divorce her. Guy is furious and loses control of himself, grabbing her by the shoulders and shaking her roughly in front of her co-workers. After storming out of Miriam's office, he calls Anne and tells her that he could kill Miriam, speaking figuratively, Ironically, Miriam is killed that night, by Anthony, who follows her to an amusement park where she is meeting a friend, and murdering her whilst Guy is riding on another train. He goes to Guy's home afterwards and tells him that she is dead, demanding that Guy hold up his end of the bargain and kill Anthony's father.
Guy goes instead to Anne's family home, where the Senator tells him that Miriam has been killed. Guy pretends to be shocked but it occurs to Anne's sister Barbara that the police will consider him a suspect because he had a motive. The police do consider him a person of interest and they interview him, but they cannot confirm his alibi; he claims to have met and talked to a professor on the train but the man was so drunk that he cannot even remember their meeting. Stymied temporarily, the police start to watch Guy around the clock to make sure that he does not try to leave town.
Anthony, meanwhile, is doing a little watching of his own; he follows Guy all over town and even introduces himself to Anne, whilst atteding a party at the Morton's home. He pretends to amuse another guest by demonstrating the correct way in which to strangle someone, gently placing his hands around her neck. When he catches sight of Barbara Morton he is struck by her resemblance to Miriam and this causes him to flash back to the murder; he becomes so caught up in the memory that he actually tries to strangle his fellow party guest before blacking out and releasing his grip on her. Barbara is disquieted and tells Anne that he was looking at her whilst he was attempting to strangle their guest. Anne, too, realizes that her sister looks like Miriam and confronts Guy about his apparent new friend. Guy buckles and tells her the full story, including the details of Anthony's murderous scheme.
Anthony sends a pistol, a key to his father's house, and a floor plan of the house to Guy. Guy pretends to be going along with the plan but instead decides to warn his intended victim. When he creeps into the bedroom he finds Anthony there, and not his father. Anthony tells him that when Guy agreed to kill his father he became suspicious. He refuses the suggestion of psychiatric help and instead tells Guy that he will make him pay for the crime that Anthony committed on his behalf.
Anne tries to take matters into her own hands. She also goes to the Anthony home, intending to tell his mother that her son has committed a murder. She too finds Anthony rather than his parents, and he mentions that he has Guy's cigarette lighter in his possession, claiming that Guy asked him to look for it at the scene of the murder. As Guy suspected, Anthony's intention is to implicate him in Miriam's murder. With the police detail still trailing him, Guy plays in a tennis match at Forest Hills, but manages to lose them and heads to the amusement park to prevent Anthony from planting the lighter there and leading the police to it as "evidence".
Anthony drops the lighter in a storm drain and is delayed; he has difficulty retrieving it and by the time he arrives at the amusement park, it is around the same time as the murder occurred. A park employee recognizes him as someone who was in the park on the night of the murder and calls the police. They think the man he is referring to is Guy, who is also in the park. He and Anthony have found each other and a fight ensues on one of the carousels. The police believe Guy is trying to escape and that the man he is involved in a tussle with is trying to stop him. They fire at Guy, but miss, instead hitting the carousel operator. The ride suddenly starts to turn more quickly until it is spinning out of control. The carousel crashes when the breaks fail. Anthony is trapped underneath the machine and cannot be saved. The worker who had originally called the police tell them that he has never seen Guy before in his life, and Guy tells them that he was at the amusement park trying to prevent Anthony from planting the lighter in an attempt to implicate him in the murder. Anthony dies, and his fingers open just enough to reveal Guy's lighter in his hand.
