Oxford academic Dr David Kerridge is believed to be missing, and a decapitated body is found in the local canal. Morse investigates, despite being encumbered by a toothache. Lewis establishes that the clothes on the body belong to Kerridge, but Kerridge is actually in London. Sir Alexander Reece, Master of Beaumont College, known to Morse from his university days, tells of a bad-tempered rivalry between Kerridge and Dr Arthur Drysdale. Recently diagnosed with brain cancer and with months to live, Drysdale has gone to Rome. Kerridge had gone to London to explore the possibility of a television appearance, but the invitation turned out to be a hoax. When he returns to his London flat after the meeting, he is attacked and killed.
Lewis believes that the canal man is Nicholas Balarat, a senior civil servant and honorary fellow of Beaumont College, who had exchanged professional criticisms with Kerridge and hasn't been seen for five days. After a trip to Whitehall to enquire about Balarat, however, Morse believes the Kerridge connection is a cover. He explains to an exasperated Chief Superintendent Strange his theory that Reece had shot Balarat. Reece had nominated Balarat for the honorary fellowship, and Morse believes he expected Balarat to return the favour by recommending him as chairman in a new royal commission. When this did not occur, Morse believes, Reece exacted revenge and disposed of the body at Thrupp to implicate Kerridge.
When the head of the canal man is found and is confirmed to be that of Balarat, the bullet wound in his skull supports Morse's theory that there are two murderers to be found. Before Morse can find any evidence of this, Reece himself is shot dead in his lodgings. Lewis had learned from a college scout that, not only did Drysdale have his disagreements with Kerridge, but also that Drysdale's wife had run off with Balarat.
Soon Morse and Lewis catch up with Dysdale, who had not gone to Rome, and he confesses to the killings. Morse suspects that Dysdale's brain cancer removed any inhibition against settled old scores. Dysdale admits killing Balarat for his betrayal and Reece for blocking him from a prestigious academic post. But he claims to have failed in murdering Kerridge, who actually convinced Dysdale that he hadn't spoken against his receiving the position. Dysdale and Morse conclude that Reece must have killed Kerridge, as Morse originally suspected, since he knew about Reece's dishonesty and fear of being exposed.
Based on Colin Dexter's novel The Riddle of the Third Mile.