The exact origin of the 3x+1 problem is obscure. It has circulated by word of mouth in the mathematical community for many years. The problem is traditionally credited to Lothar Collatz, at the University of Hamburg. In his student days in the 1930's, stimulated by the lectures of Edmund Landau, Oskar Perron, and Issai Schur, he became interested in number-theoretic functions. His interest in graph theory led him to the idea of representing such number-theoretic functions as directed graphs, and questions about the structure of such graphs are tied to the behavior of iterates of such functions [25]. In his notebook dated July 1, 1932, he considered the function

which gives rise to a permutation P of the natural numbers

He posed the problem of determining the cycle structure of P, and asked in particular whether or not the cycle of this permutation containing 8 is finite or infinite , i.e., whether or not the iterates (8) remain bounded or are unbounded [24]. I will call the study of the iterates of the original Collatz problem. Although Collatz never published any of his iteration problems, he circulated them at the International Congress of Mathematicians in 1950 in Cambridge, Massachusetts, and eventually the original Collatz problem appeared in print ([9], [47], [62]). His original question concerning (8) has never been answered; the cycle it belongs to is believed to be infinite. Whatever its exact origins, the problem was certainly known to the mathematical community by the early 1950's; it was discovered in 1952 by B. Thwaites [72].