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Zuckerman competed in seven U.S. Chess Championships (1965, 1966, 1968, 1969, 1974, 1977 and 1978), his best result being a tie for fourth place with William Addison in 1965. He served as a member of the U.S. team in the World Student Team Championships of 1964, 1967 and 1969. At Brooklyn College, Zuckerman was a prominent player, along with Raymond Weinstein, on its national champion college chess team.
For more than forty years, Zuckerman has been a well-known authority on chess openings. At the 1959 U.S. Open Chess Championship in Omaha, Nebraska, Grandmaster Arthur Bisguier, who won the tournament, often asked Zuckerman, then a Class B player, what opening to play and then followed his advice. Zuckerman became known as "Zook the Book" or "Zuckerbook" because he knew more about opening theory than could be found in any book.
Zuckerman is 22 days younger than Bobby Fischer, and sometimes jested that when he got as old as Fischer he would be just as strong. Zuckerman was one of Fischer's few close friends, and taught him much about the Sicilian Defence and other openings. Fischer used Zuckerman's advice successfully; see the 1962 game Bilek-Fischer for example.
Tournament organizers knew Zuckerman's games frequently resulted in short draws, for which reason he gradually stopped receiving invitations to international tournaments.
Zuckerman was a strong blitz chess player, displaying the same mastery of openings, and winning, rather than drawing as draws are much less common in blitz chess.
Zuckerman has not played in a serious open chess tournament since 1990. He currently lives in Brooklyn Heights, New York.