  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 constraint-based algorithms build on reductions to
$\mathrm{SAT}$ and $\mathrm{IP}$ 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 $k \le 5$, construct puzzles of small width that are
larger than previous work, and improve the upper bounds on strong USP size for
$k \le 12$. Although our work only deals with puzzles of small-constant width,
the strong USPs we find imply matrix multiplication algorithms that run in
$O(n^\omega)$ time with exponent $\omega \le 2.66$. While our algorithms do not
beat the fastest algorithms, our work provides evidence and, perhaps, a path to
finding families of strong USPs that imply matrix multiplication algorithms
that are more efficient than those currently known.
