375. Guess Number Higher or Lower II

Medium

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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 = 10Output:16Explanation: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 = 1Output:0Explanation:There is only one possible number, so you can guess 1 and not have to pay anything.

**Example 3:**

Input:n = 2Output:1Explanation: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`

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