DNA sequences that position nucleosomes are of increasing interest because of their relationship to gene regulation in vivo and because of their utility in studies of nucleosome structure and function in vitro. However, at present our understanding of the rules for DNA sequence-directed nucleosome positioning is fragmentary, and existing positioning sequences have many limitations. We carried out a SELEX experiment starting with a large pool of chemically synthetic random. DNA molecules to identify those individuals having the highest affinity for histone octamer. A set of highest-affinity molecules were selected, cloned, and sequenced, their affinities (free energies) for histone octamer in nucleosome reconstitution measured, and their ability to position nucleosomes in vitro assessed by native gel electrophoresis. The selected sequences have higher affinity than previously known natural or non-natural sequences, and have a correspondingly strong nucleosome positioning ability. A variety of analyses including Fourier transform, real-space correlation, and direct counting computations were carried out to assess non-random features in the selected sequences. The results reveal sequence rules that were already identified in earlier studies of natural nucleosomal DNA, together with a large set of new rules having even stronger statistical significance. Possible physical origins of the selected molecules' high affinities are discussed. The sequences isolated in this study should prove valuable for studies of chromatin structure and function in vitro and, potentially, for studies in vivo.