Arlena, known for sunbathing, is found face down in the sun, dead. Poirot collects alibis. Linda drops a parcel of candles when Christine asks her to Gull Cove. Arlena paddles to Pixy Cove for a rendezvous. Both Kenneth and Patrick look for her. Patrick asks Emily to join her daily row. He finds a Arlena's limp body lying face down, arms outstretched. He stays with the body while Emily fetches the doctor who concludes it is death by strangulation, likely a male.
Police questions the suspects. Kenneth was heard typing letters responding to figures in previous mail. Linda lies and says she was fond of her stepmother. She and Christine went to Gull Cove at 10:30 and didn't return until 11:45. The Gardeners were with Poirot the entire time. Emily and Patrick saw Rosamund reading at Sunny Edge. Reverend Lane and Major Barry went out. Sir Horace Blatt spent the morning sailing. Christine Rosamund, Kenneth and Mr. Gardener went to play tennis at noon. Earlier in the day, Miss Brewster narrowly missed being hit on the head with a bottle tossed from a window. Someone ran a bath at noon but no one is admitting to it.
At Pixy Cove, Poirot finds a new pair of scissors, a pipe fragment and heroin. Poirot also smells a perfume only used by Arlena and Rosamund. Poirot invites everyone on a picnic to test their vertigo: Christine, who claims she has vertigo, easily traverses the bridge. Linda overdoses on six sleeping pills, and almost dies. Linda admits to the murder, but Poirot finds her library book and realizes she thinks that piercing a voodoo doll qualifies as murder. Christine made her sleeping pills availabe and Linda took them.
Poirot investigates any local strangulations and discovers that Alice Corrigan was strangled. Her husband, Edward, was too far away to have committed the crime. Police identify Patrick Redfern as Edward Corrigan. They identify the deceased as Christine Redfern, then known as Christine Deverill, Patrick's true love. Patrick simply used Arlena for her money. While Arlena did not suspect anything, if her husband discovered she emptied their bank account, he would be suspicious. Patrick and Christine decided to get rid of her.
Patrick told Arlena to meet him that fateful day at a cave. Christine set Linda's watch ahead by 20 minutes to give her an alibi. Christine put on some tan makeup and pretended to be Arlena. Emily was fooled by her. Christine returned to her hotle room to wash off the suntan makeup, the bath that no one had admitted to. She threw the empty bottle out the window, the one that narrowly missed Miss Brewster. Meanwhile, Patrick called the unsuspecting Arlena out of the cave and strangled her.
Rosamund gives up her career to marry Kenneth and Linda ends up with a loving stepmother.