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.
