Android Unlock Patterns - Solution & Explanation

Problem Overview: Count how many valid unlock patterns can be formed on the Android 3x3 lock screen using lengths between m and n. A pattern cannot revisit a key, and moves that jump over another key are only allowed if that intermediate key has already been used.

Approach 1: Brute Force Backtracking (O(9!))

The straightforward strategy generates every possible path on the 3×3 grid using depth‑first search. Start from each digit (1–9), mark it visited, and recursively try the remaining digits while respecting the Android rule: if a move crosses another key (for example 1 → 3 crossing 2), that intermediate key must already be visited. A small lookup table called skip stores these required intermediate nodes. The search continues until the path length reaches n, counting patterns whose length is at least m. Time complexity is O(9!) in the worst case because permutations of keys are explored, and space complexity is O(9) for the recursion stack and visited array.

Approach 2: Symmetry Optimized Backtracking (O(9!))

The grid has strong symmetry. Corners (1,3,7,9) behave the same, edges (2,4,6,8) behave the same, and the center (5) is unique. Instead of running DFS from all nine digits, run it once from a representative of each group and multiply the result. For example, compute patterns starting from 1 and multiply by four for the corners. Do the same for 2 multiplied by four for edges, and handle 5 once. The same skip matrix enforces the rule about crossing intermediate keys. This reduces redundant exploration while keeping the same theoretical complexity O(9!), with space O(9). This approach is what most production solutions use because it cuts the constant factor dramatically.

Approach 3: Bitmask + DP State Exploration (O(9 * 2^9))

A more algorithmic perspective models the problem as state transitions. Each state is defined by the current key and a bitmask representing visited keys. From a state (node, mask), iterate over all possible next digits and check whether the move is valid using the same intermediate-key rule. Memoizing results for identical states avoids recomputation. The number of states is bounded by 9 * 2^9, giving time complexity around O(9 * 2^9) and space complexity O(9 * 2^9). This version connects naturally to dynamic programming and bitmask techniques.

Recommended for interviews: Symmetry‑optimized backtracking is the expected solution. It demonstrates control of DFS, constraint handling, and pruning using problem structure. Interviewers like seeing the skip matrix plus the symmetry observation because it reduces work from nine starting searches to three. Mentioning the bitmask state idea also shows familiarity with backtracking and DP tradeoffs.