I stared at this for longer than I'd like to admit.
Model the keypad as a graph where each key is a node and knight moves define edges. Use dynamic programming to count the number of ways to reach each key for each digit length, starting from all keys for length 1 and iterating up to N.
Pro tip: Clarify the keypad layout (e.g., 3x4 grid with 0 at bottom center) and discuss how the solution can be optimized using matrix exponentiation for very large N, showing awareness of scalability.
Confirm the keypad layout, definition of a knight move, and whether leading zeros are allowed. Ask about constraints on N (e.g., N up to 10^9) to determine if optimization is needed.
Represent each key as a node and precompute all valid knight moves from each key. This forms a directed graph where edges represent possible transitions.
Let dp[i][k] be the number of sequences of length i ending at key k. Base case: dp[1][k] = 1 for all k. Transition: dp[i][k] = sum of dp[i-1][j] for all j that can move to k via a knight move.
Iterate from length 2 to N, updating dp for each key. Sum dp[N][k] over all k for the answer. For large N, use matrix exponentiation on the transition matrix to achieve O(log N) time.
Discuss time and space complexity: O(N * 10) for DP, or O(10^3 log N) for matrix exponentiation. Mention that the graph is sparse, so transitions are constant time per key.
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