375. Guess Number Higher or Lower II

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Problem

We are playing the Guessing Game. The game will work as follows:

• I pick a number between 1 and n.
• You guess a number.
• If you guess the right number, you win the game.
• If you guess the wrong number, then I will tell you whether the number I picked is higher or lower, and you will continue guessing.
• Every time you guess a wrong number x, you will pay x dollars. If you run out of money, you lose the game. Given a particular n, return the minimum amount of money you need to guarantee a win regardless of what number I pick.

Example 1:

Input: n = 10
Output: 16
Explanation: The winning strategy is as follows:

- The range is [1,10]. Guess 7. - If this is my number, your total is $0. Otherwise, you pay $7. - If my number is higher, the range is [8,10]. Guess 9. - If this is my number, your total is $7. Otherwise, you pay $9. - If my number is higher, it must be 10. Guess 10. Your total is $7 + $9 = $16. - If my number is lower, it must be 8. Guess 8. Your total is $7 + $9 = $16. - If my number is lower, the range is [1,6]. Guess 3. - If this is my number, your total is $7. Otherwise, you pay $3. - If my number is higher, the range is [4,6]. Guess 5. - If this is my number, your total is $7 + $3 = $10. Otherwise, you pay $5. - If my number is higher, it must be 6. Guess 6. Your total is $7 + $3 + $5 = $15. - If my number is lower, it must be 4. Guess 4. Your total is $7 + $3 + $5 = $15. - If my number is lower, the range is [1,2]. Guess 1. - If this is my number, your total is $7 + $3 = $10. Otherwise, you pay $1. - If my number is higher, it must be 2. Guess 2. Your total is $7 + $3 + $1 = $11.
  The worst case in all these scenarios is that you pay $16. Hence, you only need $16 to guarantee a win.

Example 2:

Input: n = 1
Output: 0
Explanation: There is only one possible number, so you can guess 1 and not have to pay anything.

Example 3:

Input: n = 2
Output: 1
Explanation: There are two possible numbers, 1 and 2.
- Guess 1. - If this is my number, your total is $0. Otherwise, you pay $1. - If my number is higher, it must be 2. Guess 2. Your total is $1.
  The worst case is that you pay $1.

Code

class Solution {
    int[][] dp;
    public int getMoneyAmount(int n) {
        dp = new int[n + 1][n + 1];
        return helper(1, n);
    }

    private int helper(int i, int j){
        if(i >= j) return 0;
        if(dp[i][j] != 0) return dp[i][j];

        int res = Integer.MAX_VALUE;
        for(int x = i; x <= j; x++){
            res = Math.min(res, Math.max(helper(i, x - 1), helper(x + 1, j)) + x);
        }

        dp[i][j] = res;
        return res;
    }
}