We give a direct and explicit arithmetic description of the behavior of theCollatz map on odd integers. All odd steps are shown to pass through a single affine progression beforedivision by powers of two. An exhaustive table describes how this progression decomposes under repeateddivision by 2 and returns to odd values. No probabilistic, asymptotic, or heuristic arguments are used.
Daniel Augusto Jorge Zafaranich (Sun,) studied this question.