Discuss how the twentieth-century American mystery story evolved in a very different direction than the British version.

The British mystery formula was definitively established by Sir Arthur Conan Doyle, whose Sherlock Holmes became the pattern for the sophisticated, brilliant detective. Although Conan Doyle wrote at the end of the 1800s and in the early 1900s, the genre he began carried on into the twentieth century. In the pattern established by Conan Doyle, a detective with extraordinary reasoning powers investigates a crime (usually a murder) that has occurred within a closed setting (such as an isolated country manor) with a finite circle of suspects. Clues accumulate with many red herrings as it becomes obvious that several people had the means, motive, and/or opportunity to have performed the crime. The detective's loyal sidekick, the "Watson," often reveals the clues to the reader but doesn't solve the crime. Only the brilliant detective is able to put everything together, and he explains how he solved the crime at the end of the story. This formula was taken up by many British authors, including the renowned Agatha Christie.
The American detective novel split off in the 1920s and added its own unique flavor to the mystery genre. Cheap, sensational magazines known as "pulps" published gritty stories that featured hard-boiled detectives who navigated the era of prohibition and gangsters. The setting was often urban rather than rural, and the detective had to find the suspect among a huge population of people by following clues from one end of town to the other, sometimes at breakneck speed. The American detective was not the refined, cerebral sleuth of the British mystery but a tough "private eye" who doesn't fit in with high society and may be slightly shady himself. Dashiell Hammett was one of the originators of this genre.
In general, then, the British mystery features a sophisticated and brilliant detective operating in a closed, isolated setting while the American version features a hard-nosed private eye who pursues the suspected criminal through the streets and venues of an urban center.