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# 375. Guess Number Higher or Lower II

## Description

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

1. I pick a number between 1 and n.
2. You guess a number.
3. If you guess the right number, you win the game.
4. 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.
5. 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.

Constraints:

• 1 <= n <= 200