Tommy and Tuppence's latest client is a personable, rich but slightly dense young man called Mr. Montgomery Jones. He has met an Australian girl called Una Drake who he has fallen violently in love with. Talking of their mutual love of detective stories, she has made a bet with him that he cannot break an alibi she has set up for herself. She has agreed that if he wins he can ask her for anything he likes and it is his intention to ask her for her hand in marriage. Knowing full well that he is not known for his intelligence, Montgomery Jones asks the Beresfords to take on the task for him.
The alibi is that on the previous Tuesday Miss Drake dined at a Soho restaurant, saw a West End show and then had supper with a Mr. le Marchant at the Savoy Hotel. At the same time, she also stayed the night at the Castle Hotel in Torquay and returned to London the next morning. Armed with a photograph of Miss Drake and the knowledge of the methods of Inspector French, Tommy and Tuppence interview Mr. le Marchant who confirms that he was with the young lady for part of the evening in question. He also states that Miss Drake made an oblique comment to being in Devonshire at the same time, which he considers strange since a friend of his was at the Castle Hotel and did indeed think that he saw her there. The two then investigate the Soho restaurant (where a positive identification of the girl is not forthcoming) and then travel to Torquay where they find plenty of evidence that Miss Drake was in the hotel throughout the night and also travelled to the resort at a time compatible with her also being in London. Their final questioning is back in London of some people who saw her at the Savoy and also of her flatmate and charwoman who both attest that she spent the night in her own bed.
Working all evening on trying to puzzle out the problem, the Beresfords confess that they are stumped. Sleeping on the problem, the next morning Tuppence awakes to a flash on inspiration and sends off a cable to test her idea. Later that day she returns to their office with the solution – Una has a twin sister who, the cable to Australia has confirmed – arrived in England the day before the events in question. The fun-loving sisters thought the stunt would be an amusing joke to play on Montgomery Jones.