The story concerns Wolfe being hired to investigate documents missing from a million-dollar prize contest for a perfume company, with the title a reference to the deadline for winning entries: postmarked before midnight on the specified date.

Dahlmann's wallet was also stolen, and inside were the final answers for a series of cryptic poetic riddles run as part of a promotional competition for Pour Amour, a brand of perfume designed by one of LBA's clients.

The executives, however, do not want Wolfe to investigate the murder but to find out who stole the wallet before the contest deadline—midnight of the nineteenth of April, exactly one week later—in order to ensure the integrity of proceedings and restore their reputation.

Wolfe dispatches Archie Goodwin to secure a copy of the final riddles and their answers for Wolfe's reference, and proceeds to interview each of the contestants: Gertrude Frazee, the leader of an anti-cosmetics women's group who has been using her members to find the answers for the riddles to try and embarrass the cosmetics industry; Carol Wheelock, a housewife who wants the prize money to secure a better life for her family; Harold Rollins, a condescending academic who entered the competition as part of an intellectual exercise; Susan Tescher, a magazine editor who wants to do a profile on Wolfe himself; and Philip Younger, a retiree seeking to recover a fortune he lost during the Great Depression.

As the deadline nears, the LBA executives begin to panic and lash out, resulting in a contradictory sequence where Wolfe is fired and then rehired within a span of minutes.

His suspicions aroused, Wolfe summons the major players to his office and claims he will reveal the identity of the thief, and in doing so provide the police with vital information to help them identify the murderer.

He, along with Archie and Saul Panzer, travels to the offices of LBA and inspects a display of products from their clients, discovering a bottle of cyanide that he suspects was used to murder Assa; this confirms in Wolfe's mind the guilt of one of the executives.