4 Keys Keyboard

Problem Statement

In this problem, you are provided with a simulated special keyboard that has keys to print the character 'A', select the screen, copy the selection, and paste the copied content. The challenge is to determine the maximum number of 'A's one can generate on the screen using a sequence of no more than n key presses. The different strategies blending these keys effectively to maximize output vary with the value of n, requiring a thoughtful approach to balance direct printing versus operations that manipulate multiple characters, such as copying and pasting.

Examples

Example 1

Input:

n = 3

Output:

3

Explanation:

We can at most get 3 A's on screen by pressing the following key sequence:
A, A, A

Example 2

Input:

n = 7

Output:

9

Explanation:

We can at most get 9 A's on screen by pressing following key sequence:
A, A, A, Ctrl A, Ctrl C, Ctrl V, Ctrl V

Constraints

• 1 <= n <= 50

Approach and Intuition

The intuitive approach to this problem involves understanding the trade-offs between simply pressing 'A' multiple times and using the combination of select, copy, and paste operations:

1. For a small number of presses (typically less than 4), the best approach is to directly press 'A' as many times as allowable by n, since using copy-paste operations wouldn't be efficient or wouldn't be possible at all with less than three presses (one for copying and at least two for selecting and pasting).

2. As n increases, it becomes viable to use a combination of the keys to multiply the 'A's produced:

   • Initially printing a few 'A's.
   • Using Ctrl-A and Ctrl-C to select and copy all 'A's.
   • Using Ctrl-V multiple times to double, triple, or further multiply the number of 'A's.

3. The approach effectively shifts from direct typing to an operational strategy as the value of n increases, reflecting a point where the operations (select, copy, and paste) provide a multiplicative effect beyond what direct typing would achieve.

4. Key sequences will generally start to include one cycle of 'Ctrl-A' and 'Ctrl-C' followed by as many 'Ctrl-V' presses as can fit within the remaining key presses after initial 'A' typings. This acknowledges that each paste operation post copying doubles the count of 'A's already present on the screen, maximizing output when planned correctly.

By evaluating these points for different values of n, we can construct a dynamic approach to calculate the maximum 'A's for any given n. This involves testing smaller groups of 'A' prints and 'Ctrl-V' operations to see what combination fits 'n' presses while producing the most output. The solution to this kind of problem involves both dynamic programming and insightful breakdowns of operations to maximize the 'A's generated given the transferable nature of the copying operation.

Solutions

class Solution {
public:
    int maximumAchievable(int steps) {
        vector<int> results(steps + 1);
        iota(results.begin(), results.end(), 0);
        for (int idx = 0; idx <= steps - 3; idx++) {
            for (int jdx = idx + 3; jdx <= min(steps, idx + 6); jdx++) {
                results[jdx] = max(results[jdx], (jdx - idx - 1) * results[idx]);
            }
        }
        return results[steps];
    }
};