One, Two, Buckle My Shoe

Whilst attending an appointment with his dentist, Dr Henry Morley, Hercule Poirot learns that Morley's secretary, Gladys Nevill, is absent. Outside the surgery, Poirot then encounters former actress Mabelle Sainsbury Seale exiting a taxi and retrieves a lost shoe buckle for her.

Later that day, Japp informs Poirot that Morley has been found dead, killed by a gunshot wound to the head, and that Morley was holding a gun. Upon investigating, Poirot learns that between his appointment and Morley's death, the dentist had seen three patients – Mabelle Sainsbury Seale, whom Poirot met earlier; prominent financier Alistair Blunt; and a Greek gentleman named Amberiotis. Howard Raikes, an American left-wing activist who wishes to marry Blunt's niece Jane Olivera, also visited the surgery but left without seeing Morley's partner Dr Reilly.

Subsequently, Amberiotis is found dead, killed by an overdose of anaesthetic. Tying the two deaths together, the police conclude that Morley accidentally injected the overdose during Amberiotis's appointment and committed suicide upon realising the mistake. Poirot, however, rejects this view.

Morley's secretary, Nevill, reappears and reveals that the reason for he absence was a telegram calling her away but that she found the content of the telegram to ultimately prove false. Poirot recalls that Morley, who disliked Nevill's boyfriend Frank Carter, had believed Carter sent the telegram as a ruse to enable he and Nevill to spend time together.

Meanwhile, Mabelle Sainsbury Seale disappears after speaking with the police. A month later, a badly disfigured body is found in the apartment of a Mrs Albert Chapman, a woman who has also disappeared. Poirot notes the dullness of the buckled shoes on the body, which is attired in clothes similar to Mabelle's. However, dental records reveal the body to be that of Mrs Chapman.

Blunt invites Poirot to his country house, where he meets Blunt's cousin Helen and niece Jane. Later, a shot is fired at Blunt; the culprit is apparently Frank Carter, Blunt's new assistant gardener, as Raikes finds Carter holding a gun of the same make as the gun that killed Morley. Furthermore, Morley's maid Agnes Fletcher admits she saw Carter on the office's back stairs, before Morley's death. Poirot presses Carter, now held by police, for the truth. Carter admits he did go to confront Morley, and saw two men leave the office by the back stair; however, when Carter himself entered the office, Morley was already dead. Carter also claims that he found the gun in the shrubbery after hearing the shot, and never actually fired at Blunt.

Poirot confronts Blunt and accuses him of the murders. Blunt's "cousin" Helen Montressor is really his first wife, Gerda; the real Helen had died years before. Gerda and Mabelle had been in the same repertory theatre company in India when Blunt secretly married Gerda. He had not divorced Gerda before he married his now-deceased and socially appropriate second wife, Rebecca Arnholt. If his bigamy were exposed, he would be disgraced, lose the fortune he inherited from Arnholt, and be removed from his position of power. When Mabelle recognised Blunt years later, she knew nothing of his second marriage. Later, Mabelle met Amberiotis, a casual acquaintance, and mentioned encountering her old friend's husband. Amberiotis decided to blackmail Blunt by threatening to expose the bigamy.

The day of Poirot's appointment with Morley, Gerda and Blunt decoyed Morley's secretary away by the false telegram. Gerda had already invited Mabelle to an apartment she had secured under the alias of Mrs Chapman, poisoned her, and stole her identity. Gerda, as "Mabelle", then visited the dentist's waiting room, secure in her deception since Gladys was away. After Morley had taken care of Blunt's teeth, Blunt shot him, took Morley's place, and buzzed the page so he could send "Mabelle" up as the next patient. The two moved the body into a side room, and Gerda changed Mabelle's records to become those of Mrs Chapman and vice versa, to confuse the police. Gerda, as "Mabelle", left the office by the back staircase most patients usually took; Blunt, as "Morley", then saw Amberiotis (who had never met Morley or Blunt in person before), and injected his gum with the overdose. Once Amberiotis left, Blunt moved Morley's body back into the surgery, set up the "suicide" scene, and left, also via the back stair, unaware that he and Amberiotis had been seen leaving by Carter. Gerda wore new shoes when impersonating Mabelle, as she could not fit into Mabelle's larger shoes; she left Mabelle's own old shoes on the dead body, a detail that Poirot noticed. Later, as "Helen", Gerda fired a shot in the vague direction of Blunt and left the gun for Carter to find, setting him up as their scapegoat.

Unrepentant, Blunt makes it clear he expects Poirot to cover for him because of his political importance. Poirot refuses; though he believes Blunt's policies are better for England than those of Blunt's opponents, he cannot leave the untimely deaths of three human beings unavenged. As Gerda and Blunt are arrested, Poirot meets Raikes and Olivera and tells them to enjoy their life together, asking that they allow freedom and pity within it.