A hiker finds a dead woman on the North York Moors. The victim had been strangled and is identified as Alice Ruber. Belgian detective Hercule Poirot (Peter Ustinov) is asked to examine a diamond ring belonging to millionaire industrialist Sir Horace Blatt (Colin Blakely). Poirot declares it a fake and assured that Sir Horace would have had given a real diamond to his mistress and that Arlena Stuart Marshall (Diana Rigg) had returned the fake, Poirot agrees to meet Sir Horace at an exclusive Adriatic Sea island and confront her. The hotel is the former summer palace of the reigning King of Tyrania, now owned by Daphne Castle (Maggie Smith), who had received the palace “for services rendered”.
Sir Horace’s former mistress is an actress on holiday with her husband Kenneth (Denis Quilley). Arlena is emotionally abusive to his daughter, Linda (Emily Hone) and flirts with Patrick Redfern (Nicholas Clay) who is married to Christine (Jane Birkin). Patrick is on the island only because Arlena arranged it. Kenneth turns to his old friend, Daphne, who reviles the way that Arlena treats both him and Linda. Arlena has also caused financial problems for Odell and Myra Gardener (James Mason and Sylvia Miles) by walking out of a major play and refusing another. Writer Rex Brewster (Roddy McDowall) has already spent the royalties advanced to him for a tell-all biography of Arlena, but she refuses to give him a release. Early on the third morning, Arlena takes a paddle-boat to Ladder Bay. Patrick and Myra go for a boat trip around the island and see a body lying motionless on the beach. Patrick approaches the body and recognizes Arlena, announcing that she has been strangled. Poirot must determine which of his eight fellow guests—or Daphne—is the murderer.
Daphne had heard Kenneth in his room typing at the time of the murder, and Christine was with Linda at Gull Cove and did not leave until 11:55 for a 12:30 tennis match. Sir Horace argued with Arlena about the diamond at Ladder Bay at 11:30, which is confirmed by his yacht crew and by Daphne. Arlena kept the diamond, promising an explanation that evening, and Poirot finds the fake jewel nearby in a grotto. Patrick left at 11:30 with Myra, seeing Sir Horace’s yacht coming, and hearing the noon cannon. Rex met Linda entering Gull Cove at 12:00 and reports that a bottle flung from the top of a cliff nearly hit him. Odell was seen reading by Daphne and her staff. He claims low water pressure hindered his 12:15 wash before tennis, but nobody admits to bathing at that time.
Assembling the suspects together, Poirot accuses Christine and Patrick of the crime: Christine knocked out Arlena and hid her in the nearby grotto, and Patrick strangled the helpless Arlena later. Christine posed as Arlena with makeup to simulate a suntan, Arlena’s swimsuit and large red hat, to be purposely mis-identified by Patrick in Myra’s presence. But Poirot had smelled Arlena’s perfume in the grotto. Christine set Linda’s watch twenty minutes fast, suggested a swim cap to muffle the noon cannon, and corrected the watch afterward. She tossed out the lotion bottle, almost hitting Rex, and bathed off her tan, thus depriving the leaky hotel water system of pressure. Poirot suspects that Patrick switched Sir Horace’s jewel with a copy and that Patrick and Christine killed Arlena to protect the theft. The Redferns scoff at the detective’s accusations, as he has no real proof.
On leaving the hotel, Patrick pays by cheque, signing the “R” in “Redfern” in a distinctive way that Poirot recognises as the same way “Felix Ruber”, husband of the Yorkshire moor victim, signed his name. The hiker that found the body had been Christine, establishing Patrick’s alibi. Poirot knows photos from the British police will show Patrick and Felix to be the same person. Patrick puts a pipe in his mouth that has never been lit during his stay; Poirot empties the pipe bowl to reveal the genuine diamond. Patrick Redfern sucker punches Poirot into unconsciousness. The closing scene shows Daphne feeding the now conscious Poirot and informing him that the king is awarding him the Order of St Goodwin The Inquisitive, First Class, as Kenneth and Linda look on smiling. Meanwhile, several members of Daphne’s staff are shown holding the Redferns prisoner on the island’s shuttle boat on the way to the mainland while Blatt, Brewster, and the Gardners gleefully taunt the murderous couple with a champagne toast from Blatt’s yacht.