Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras . In subsequent work with Kleinberg and Szegedy, they connected this to the search for combinatorial objects called strong uniquely solvable puzzles (strong USPs) . We begin a systematic computer-aided search for these objects. We develop and implement algorithms based on reductions to SAT and IP to verify that puzzles are strong USPs and to search for large strong USPs. We produce tight bounds on the maximum size of a strong USP for width k<6, and construct puzzles of small width that are larger than previous work. Although our work only deals with puzzles of small-constant width and does not produce a new, faster matrix multiplication algorithm, we provide evidence that there exist families of strong USPs that imply matrix multiplication algorithms that are more efficient than those currently known.