The 'passing through' constraint is where I stumbled.
Model the keypad as a graph where edges represent valid moves, including special 'skip' edges that require an intermediate key to be visited first. Use DFS with backtracking to explore all valid patterns of length between m and n, leveraging symmetry to reduce redundant computations.
Pro tip: Precompute the skip conditions and use symmetry: patterns starting from 1, 3, 7, 9 are equivalent, as are 2, 4, 6, 8, and 5 is unique. This reduces the search space by a factor of 8 and shows optimization awareness.
Represent each key as a node and define valid moves between keys. For moves that pass over an intermediate key, store the required intermediate key.
Create a 2D array 'skip' where skip[i][j] is the key that must be visited before moving from i to j, or 0 if no intermediate key is required.
Starting from each key, recursively explore all valid next keys that are unvisited and whose skip condition (if any) is satisfied. Track the current pattern length.
At each recursion step, if the current length is between m and n, increment the count. Continue until length n.
Compute patterns for one key from each symmetry group (corner, edge, center) and multiply by the group size to avoid redundant work.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.