375. Guess Number Higher or Lower II
Description
We are playing the Guessing Game. The game will work as follows:
 I pick a number between
1
andn
.  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 payx
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
Solutions
Solution 1
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