The $2 \times 3$ sliding puzzle board is serialized into a 6-character string. Blank tile '0' swaps with precomputed adjacent indices. The state space has at most $6! / 2 = 360$ reachable permutations.
DIR = [[1,3],[0,2,4],[1,5],[0,4],[1,3,5],[2,4]] guides all 0-swaps. The target is reached when string equals "123450".
"A 2D $2 \times 3$ grid is inconvenient to hash. We flatten the board into a 6-character string like '123405'. This enables $O(1)$ string hashing and set lookups."
"The blank tile '0' only swaps with its cardinal neighbors. We hardcode DIR = {{1,3},{0,2,4},{1,5},{0,4},{1,3,5},{2,4}} to avoid 2D coordinate conversion inside the inner loop."
"There are $6! = 720$ total permutations of 6 digits. Due to parity of inversions, exactly half ($720 / 2 = 360$) are reachable. Thus, BFS visits at most 360 states before concluding."
"Because each swap costs 1 move, BFS guarantees that the first time we encounter '123450', it is the minimal move sequence."