This paper presents new results on an innovative approach for solving satisfiability problems (SAT), that is, creating a logic circuit that is specialized to solve each problem instance on Field Programmable Gate Arrays (FPGAs). This approach has become feasible due to recent advances in Reconfigurable Computing, and has opened up an exciting new research field in algorithm design. We have developed an algorithm that is suitable for a logic circuit implementation. This algorithm is basically equivalent to the Davis-Putnam procedure with Experimental Unit Propagation. The algorithm requires fewer hardware resources than previous approaches. Simulation results show that this method can solve a hard random 3-SAT problem with 400 variables within 1.6 minutes at a clock rate of 10MHz. Faster speeds can be obtained by increasing the clock rate. Furthermore, we have actually implemented a 128-variable, 256-clause problem instance on FPGAs.