I\'ve read in one of my AI books that popular algorithms (A-Star, Dijkstra) for path-finding in simulation or games is also used to solve the well-known \"15-puzzle\".
A quick Google search turns up a couple papers that cover this in some detail: one on Parallel Combinatorial Search, and one on External-Memory Graph Search
General rule of thumb when it comes to algorithmic problems: someone has likely done it before you, and published their findings.