An afternoon-long tutorial followed, teaching him, by Mr. Sondheim’s account, more about the craft than most songwriters learn in a lifetime. Hammerstein laid out a path of writing exercises for him: Adapt a good play into a musical; adapt a flawed play into a musical; adapt a story from another medium into a musical; and, finally, write a musical from your own original story. This the young Mr. Sondheim did, a project that carried him through his graduation from Williams College in Massachusetts, where he complemented his theater work with serious composition study under Robert Barrow, an intellectually rigorous specialist in harmony, from whom Mr. Sondheim gleaned the lesson, as he put it, “that art is work and not inspiration, that invention comes with craft.” Mr. Sondheim would later study independently with Milton Babbitt, the avant-garde composer.

Mr. Sondheim’s first professional show business job was not in the theater at all; through the agency representing Hammerstein, he was hired to write for a 1950s television comedy, “Topper,” about a fussbudget banker haunted by a pair of urbane ghosts. (Much later, Mr. Sondheim wrote a whodunit film script, “The Last of Sheila,” with the actor Anthony Perkins; it was produced in 1973 and directed by Herbert Ross.) By the ’50s he had become a connoisseur of word games and puzzles, and an inventor of elaborate games. From 1968 to 1969, he created cryptic crosswords for New York magazine.

His affinity for theatrical misdirection and mystery was acknowledged by his friend, the playwright Anthony Shaffer, who based the cunningly vengeful cuckold in his play “Sleuth” partly on Mr. Sondheim. (The play was once tentatively titled “Who’s Afraid of Stephen Sondheim?”)

Breaking Into Broadway

Mr. Sondheim was in his early 20s when he wrote his first professional show, a musical called “Saturday Night,” which was an adaptation of “Front Porch in Flatbush,” a play by Philip G. and Julius J. Epstein. He got the job, to write both words and music, after the composer Frank Loesser turned it down. The show was scheduled to be presented in 1955, but the producer, Lemuel Ayers, died before he had completed raising the money for it, and the production came to a halt. The show was not presented until 1997, by a small company in London; it subsequently appeared in Chicago and finally had its New York premiere in 2000, Off Broadway at the Second Stage Theater.

Mr. Sondheim was loath to take either of his first Broadway gigs, “West Side Story” and “Gypsy,” because he felt he was a composer, not only a lyricist — “I enjoy writing music much more than lyrics,” he confessed in “Finishing the Hat.” But he agreed to both on the advice of Hammerstein, who told him that he would benefit from working with the likes of Bernstein; Laurents, (who wrote the book) and the director Jerome Robbins, in the first instance, and from writing for a star like Ethel Merman in the second, even though it was she who had wanted a more experienced Broadway hand, Jule Styne, as the composer.