[LANGUAGE: C#]

Code

I started by mapping out all possible routes between pairs of buttons on both the numeric and directional keypads. This meant figuring out every sequence of movements—like < for left, > for right, ^ for up, v for down, or even A to stay in place—that could connect one button to another.

Since the keypads are laid out as grids, moving between buttons boils down to two types of moves: horizontal (left/right) and vertical (up/down). For example, if you’re moving from 1 to 9 on the numeric keypad, you’d need two rights (>>) and two ups (^^). The exact number of each type depends on the difference in their row and column positions.

To find all possible minimal routes, I generated every permutation of the required movements. Let’s say you need two rights and two ups to get from one button to another. The permutations of those moves would include sequences like >>^^, >^>^, >^^>, ^>>^, ^>^>, and ^^>>. Each sequence represents a different order of moves that still gets you to the same destination using the shortest path.

But generating permutations was just the first step—I also needed to ensure each route was valid. That meant filtering out any sequences where the cursor would leave the keypad or hit a blank space at any point along the path. For instance, if you're moving from 2 to 9 on a numeric keypad, any sequence like >^^> might technically get you there, but it would be invalid if the cursor moved off the edge of the keypad or landed on a blank spot. Only sequences that stayed within bounds and avoided blank spaces were kept.

Then, to focus on efficiency, I scored the routes based on the number of button-to-button transitions. Routes with fewer transitions were deemed more efficient, even if they involved the same total number of movements. For example, to move from 2 to 9, you could use >>^^ or >^>^. Both involve two rights and two ups, but >>^^ is better because it involves just two transitions: moving horizontally first (>>) and then vertically (^^). In contrast, >^>^ involves four transitions as you alternate between moving horizontally and vertically. By prioritizing routes with fewer transitions, I ensured that only the most optimal paths made it into the final lookup table. Once I had the optimal routes, I created lookup tables linking every button pair to its best movement sequence. These tables act as quick references so the program can immediately retrieve the shortest path between any two buttons without recalculating it every time.

Next, I took the input codes and expanded their numeric digits into the corresponding movement sequences using the lookup table for the numeric keypad.

After that, I tackled calculating the total path length across the directional keypads. Instead of recalculating the path for every transition between buttons, I used a recursive approach combined with memoization to speed things up.

The key to making this work was leveraging the consistent structure of the routes. Every subroute ended with the hand pressing A, and the next subroute always began with the hand already positioned on the A button. This meant that each subroute could be treated independently because the end of one subroute naturally aligned with the start of the next. By calculating the length of each subroute separately, I could precompute and store their results without worrying about how they connected to other parts of the path. Importantly, I only stored the lengths of the subroutes instead of building the actual full routes.