My first instinct was recursion with memoization which felt fine until I noticed the hop count could go up to 5000.
Model the keypad as a graph where each digit is a node and knight moves define edges. Use dynamic programming to count sequences of length N+1 starting from the given digit, applying modulo 10^9+7 at each step. Optimize space by keeping only the previous hop's counts.
Pro tip: Clarify the keypad layout and knight move rules upfront, and mention that the two unusable corners are typically the bottom-left and bottom-right (digits * and #). Also, discuss potential optimizations like matrix exponentiation for very large N.
Confirm the keypad layout (usually 1-9, 0, with * and # as unusable corners) and that the knight starts on the given digit. Ensure you understand that a sequence of length N+1 includes the starting digit and N hops.
Map each digit to its valid knight-move neighbors on the keypad. Precompute this adjacency list to avoid recalculating moves during DP.
Let dp[hop][digit] be the number of sequences of length hop+1 ending at digit. Initialize dp[0][start] = 1. For each hop, dp[hop][d] = sum(dp[hop-1][neighbor]) for all neighbors of d, modulo 10^9+7.
Since only the previous hop's counts are needed, use two arrays (or a single array updated in place with care) to reduce space complexity from O(N*10) to O(10).
After N hops, sum the counts for all digits (or just the count for the starting digit if the sequence must end at the start? Clarify: typically sum over all possible ending digits). Return the sum modulo 10^9+7.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.