Maximum Points You Can Obtain from Cards

A row of cards each carries a point value, and you must take exactly k of them.

Constraint: In each step, you can take one card from either the beginning or the end of the row.

Goal: maximize the total points of the cards you have taken.

Example:

• nums = [1, 2, 3, 4, 5, 6, 1], k = 3
• Output: 12
• Explanation: Take [1, 2] from left and [1] from right? Sum = 4.
  Or take [1] from left and [6, 1] from right? Sum = 8.
  Or take [5, 6, 1] from right? Sum = 12. (Max)

Instead of simulating every decision (Recursion/DP), which is slow, realize that we always select k cards total. If we take i cards from the left, we must take k - i cards from the right.

The Sliding Window Approach:

1. Start: Assume we take all k cards from the beginning (left side). Calculate this current_sum.
2. Slide: In each step, we give up one card from the left and take one card from the right.
   • Subtract card[k - 1 - i] (the rightmost card of our left selection).
   • Add card[n - 1 - i] (the corresponding card from the end of the array).
3. Track: Keep track of the maximum sum found during this sliding process.

Watch how we start with the first k cards, then progressively swap a left card for a right card to find the optimal combination.