Breadth-first search explores an unweighted graph level by level and guarantees a path with the fewest edges. Dijkstra handles non-negative edge costs, while A* adds an admissible heuristic to prioritize promising states. The same state-graph model can describe puzzles as well as maps. In gothiclockpicksolver, a node can represent the positions of four to eight plates, each legal left or right input is an edge, and same-direction or opposite-direction links update multiple coordinates together. A shortest strategy can search for a minimum-move sequence, while different cost functions can prefer safer paths or fewer plate switches. In each case, use canonical state hashing, keep a visited or best-cost table, store predecessors for path reconstruction, and replay the final sequence to produce a verifiable state timeline.