Japp asks Poirot to join him at a house in Bardsley Garden Mews where a Mrs Barbara Allen shot herself the previous evening – Guy Fawkes Night – the moment of death being disguised by the noise of fireworks. Once there they find that the doctor thinks there is something strange about the death of the fine lady, a young widow. Mrs Allen was found by a housemate, Miss Jane Plenderleith, who had been away in the country the previous night. The victim was locked in her room and was shot through the head with an automatic, the weapon being found in her hand. The doctor however points out that the gun is in her right hand while the wound is above the left ear – an impossible position to shoot with the right hand. It looks as if this is a murder made to look like suicide – and by an unusually incompetent murderer with a very low estimation of the intelligence of police investigators. They interview Miss Plenderleith and find out that Mrs Allen was engaged to be married to Charles Laverton-West, an up-and-coming young MP but, although the pistol was the dead lady's, she cannot think of a reason why she should use it to commit suicide.
Japp and Poirot find further clues: the gun has been wiped clean of fingerprints and large sums of money have been withdrawn from Mrs Allen's bank account on several occasions but there is no trace of money in the house. They also learn from a neighbour that Mrs Allen had a gentleman caller the previous evening whose description doesn't tally with that of her fiancé. Feeling that Miss Plenderleith is keeping something back, they ask her about this male visitor and she suggest that it was Major Eustace – a man that Mrs Allen had known in India and whom she has seen on several occasions in the past year. She got the feeling that Mrs Allen was afraid of the man and Japp and Poirot suggest that Major Eustace was blackmailing her – an idea which meets with approval from Miss Plenderleith. Poirot points out though that it is unusual for blackmailers to kill their victims, normally it is the opposite way round. Japp, as part of his look round the house, searches a cupboard under the stairs which contains items such as umbrellas, walking sticks, tennis racquets, a set of golf clubs and a small attaché-case which Miss Plenderleith hurriedly claims is hers. The two men sense Miss Plenderleith's heightened tension.
Miss Plenderleith proves to have an impeccable alibi for the time of the death and Poirot and Japp interview Charles Laverton-West. He is stunned to find out that a murder investigation is taking place and admits that he himself has no sound alibi. They also try to see Major Eustace and hear that he has gone off to play golf. Mention of this suddenly makes Poirot see everything clearly. Managing to get hold of Eustace later on, they notice that he smokes a brand of Turkish cigarette whose stubs were found in the mews house, even though Mrs Allen smoked a different kind. They also prove that he wore a set of cufflinks, a damaged part of which was found in the room where Mrs Allen died and Japp arrests him for murder.
On a pretext, Poirot makes Japp call at the mews house and while they are there Poirot sneaks another look at the cupboard under the stairs and sees that the attaché-case has gone. As Miss Plenderleith has just come back from playing golf at Wentworth, they go there and find out that she was seen on the links with the case. Later investigations show that she was seen to throw the item into the lake there. The police retrieve it but find nothing in it. Poirot asks Japp and Miss Plenderleith to call at his flat and they tell her of Eustace's arrest. Poirot then tells her of his real conclusions. From clues concerning missing blotting paper, Poirot deduces that Mrs Allen had written a letter just before she died, which if she killed herself, would indicate a suicide note. He postulates that Miss Plenderleith came home, found her friend dead, driven to kill herself by the actions of her blackmailer and was determined to avenge her – this wasn't a murder made to look like suicide but a suicide made to look like murder and thereby entrapping the blackmailer. Miss Plenderleith placed the gun in Mrs Allen's right hand, despite the fact that she was left-handed, and the purpose of her trip to Wentworth was to hide there the dead lady's golf clubs – left-handed clubs, the attaché-case being a red herring to put the police off the trail. Convinced by Poirot that Major Eustace will be imprisoned for his other crimes, she agrees to tell the truth and save the man from the gallows.
A house party is underway at the home of Lord Charles Mayfield, a rising politician and a millionaire whose riches came from his engineering prowess. With him are Air Marshal Sir George Carrington, his wife Lady Julia and son Reggie, Mrs. Vanderlyn, who is a beautiful brunette American woman, and Mrs. Macatta, a forthright MP. Mr. Carlile, Lord Charles' secretary, joins them for dinner. The reason for the house party becomes obvious when all but Lord Mayfield and Sir George leave the dinner table, as they will discuss the plans for a new fighter aircraft that would give Britain supremacy in the air. They discuss Mrs. Vanderlyn, who is involved in espionage. Lord Charles invited her to tempt her with something big – the plans for the new fighter – to trap her once and for all.