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`