Here, the authors studied the directed assembly of the asymmetric ternary blends, composed of polystyrene-block-poly(methyl methacrylate) copolymer (PS-b-PMMA) and the corresponding PS and PMMA homopolymers, on a checkerboard chemical pattern which was fabricated by e-beam lithography, controlling the periodicity (L-S), length (D), and spacing of the exposed lines or dashed lines in the chemical pattern. The checkerboard chemical pattern, which cannot be generated with typical self-assembled block copolymer morphologies, consists of either offset, parallel, dashed lines, or alternating lines and dashed lines, and is used in the fabrication of dynamic random access memory. The degree of perfection and domain uniformity of the assembled block copolymer thin films on the complex pattern were a function of the commensurability of the volume fraction of PS in the blend (phi(S)) with the fraction of area on the pattern wet by PS (F-S), as well as the volume fraction of homopolymer in the blend (phi(H)). The best assembly occurred phi(S) and F-S were commensurate, and phi(H) was optimized for the given blend and pattern.