A young secretarial student, Margaret Bell, has died of what looks to be natural causes, and Endeavour Morse is called in to investigate. At the scene, he finds a dirty ashtray, a blanket from Beaufort College labeled with its owner's name, "Denis Bradley", and medication for her congenital heart defect. A straightforward death, according to the new chief superintendent, Bright. Bright disapproves of Endeavour, being Thursday's bagman. According to the new chief, Endeavour should be investigating the theft of money from gas meter coin boxes and a series of robberies at local post offices, leaving the murders to his more experienced superiors.
But Endeavour is certain that something is amiss in the Bell case. He learns from her boyfriend, Bradley, that neither of them smoked. Speculating that there must be another man in the picture, he learns from Margaret's friend that she was going to meet a young man the night she died, possibly Derek Clark, who works at his family's post office/grocery store.
When Endeavour questions Derek and his father Wally, he notices a pretty blond making a transaction with Derek. But Endeavour becomes more certain that Margaret's death wasn't accidental when a general practitioner, Dr. Frank Cartwright, is found dead at a riverside public bathroom. Cartwright's partner in his practice, Prentice, was Margaret's doctor. Thursday sees only coincidence where Endeavour sees a connection, and warns him to be wary of Bright. But Endeavour cannot resist showing off in front of Bright, deducing that the bicycle abandoned at the crime scene belongs to a left-handed vicar.
Cartwright was the son-in-law of Edmund Sloan, a physics professor at Beaufort. Sloan, one of the physicists who developed the nuclear bomb, has been receiving anonymous letters with the word MURDERER typed out. His daughter, Helen, is overprotective of her toddler son, who – it turns out – is her sister Pamela's biological son. When Endeavour questions the Sloan/Cartwright family about Cartwright's death, it's Pamela who reacts the worst, wracked by a seizure. Sloan reveals that Pamela (the pretty blond from the Clark's post office) is epileptic, and the widow of a man they'd never met, who died in a car crash a few years before. Helen and Cartwright are raising Bobby because the family thinks Pamela is unfit. None of them has a solid alibi. Endeavour is moved by Pamela's suffering. She tells him that if it hadn't been for Frank Cartwright, she'd still be locked up in the epileptic colony where she'd been sent away. But Pamela herself becomes Bright's favorite suspect when Endeavour's rival, Jakes, finds out that she had a record of violence at the colony.
Meanwhile, Endeavour questions Reverend Mumford about the bike, which he says was stolen from him. A periodic table indicates Mumford's interest in science, and as the two talk, Endeavour reveals that he was a cypher clerk in the Signal Corps before joining the police. As they part, Mumford shares a puzzle with Endeavour: "After beard teased, an exclamation of surprise will bring you home. Eleven letters."
When Endeavour learns that Denis Bradley, a student of Edmund Sloan's, had been typing Sloan's memoirs, he discovers that Bradley had typed the anonymous letters on Margaret's typewriter, in a passionate protest of Sloan's work on the bomb. Meanwhile, Endeavour is called to the mortuary. Margaret Bell's stomach contents have a partially dissolved amphetamine tablet.
Puzzling over a connection that everyone else refuses to see, Endeavour finds a package of amphetamines hidden at the Cartwright murder site. He is certain that Derek Clark supplied them to Margaret, and suspects that Cartwright was involved, a suspicion confirmed by Dr. Prentice, who notes that Cartwright had been ordering surplus amphetamine over the past several months. Derek denies knowing Cartwright or supplying the drugs. That night, their post office is robbed, and he and his father are found beaten and bound, two fingers severed from his father's hand when the burglars forced him to open the safe.
Worse still, Monkford is murdered. Odd, since the only suspicious experience he'd had were his stolen bike and a stolen collection of rare coins. But when the gas meter thief is caught, and Endeavour finds one of Monkford's missing coins among the loot, Endeavour is able to track coin collection thief to the gas meter burglary victim – it was he who fed the coin into the meter, and therefore he who robbed Monkford. But the thief has no regret; he thinks that Monkford had it coming because he had tried to prey on him when he was vulnerable. Endeavour then realizes that Monkford was probably visiting the bathroom for a gay assignation, and that's why he couldn't come forward about witnessing Cartwright's murder.
Seeing that Monkford's murderer had ransacked the place in search of something, and discovering that Monkford himself had served in the Signal Corps as a cryptographer, Endeavour solves Monkford's puzzle: "Breadcrumbs." Monkford had been telling Endeavour in code that he would leave clues to the identity of his murderer if Cartwright's killer were to kill him.
In this way, Endeavour solves the mystery. At Monkford's church, he unravels the mystery for Thursday, and an enraged Bright, who had already demoted him to general duties for his insubordination in continuing the investigation. Cartwright, he reveals, had found Pamela, suicidal, in London. They fell in love, and he got her pregnant. They invented the story of her dead husband, and he sent a money order to her every month, which she would cash at the Clark's post office. Eventually, Derek caught wind of the arrangement and blackmailed Cartwright for amphetamines. The killer murdered Cartwright at the drop, and Monkford witnessed the killing. Worried for his safety, Monkford concealed the killer's identity in the hymn numbers he'd posted at the church, the code corresponding to the periodic table, which spelled out the killer's identity: W CLARK. To protect his son from being discovered as the one who gave Margaret the amphetamines, and therefore killed her, Derek's father murdered Cartwright. The witness, Monkford, had to die. Clark staged the robbery and cut off his own fingers to create an alibi for himself and the son he so desperately wanted to protect.