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A switch module M with W terminals on each side is said to be universal if every set of nets satisfying the dimensional constraint (the number of nets on each side of M is at most W) is simultaneously routable through M. In this paper, we present a class of universal switch modules. Each of our switch modules has 6W switches and switch-module flexiblity three (Fs = 3). We prove that no switch module with less than 6W switches can be universal. We also compare our switch modules with those used in the Xilinx XC4000 family FPGA's and the anti-symmetric switch modules (with Fs = 3^1) suggested by [Flexibility of interconnection structures for field-programmable gate arrays]. Although these two kinds of switch modules also have Fs = 3 and 6W switches, we show that they are not universal. Based on combinatorial counting techniques, we show that each of our universal switch modules can accommodate up to 25% more routing instances, compared with the XC4000-type one of the same size. Experimental results demonstrate that our universal switch modules improve routability at the chip level. Finally, our work also provides a theoretical insight into the important observation by Rose and Brown [Flexibility of interconnection structures for field-programmable gate arrays] that Fs = 3 is often sufficient to provide high routability.