'''

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.


Constraints:

1 <= n <= 200

'''


class Solution:
    def getMoneyAmount(self, n):
        need = [[0] * (n + 1) for _ in range(n + 1)]
        for lo in range(n, 0, -1):
            for hi in range(lo + 1, n + 1):
                need[lo][hi] = min(x + max(need[lo][x - 1], need[x + 1][hi])
                                   for x in range(lo, hi))
        return need[1][n]



    def getMoneyAmount3(self, n: int) -> int:
        import functools
        @functools.lru_cache(maxsize=None)
        def get_amount(left: int, right: int) -> int:
            if left >= right:
                return 0
            return min(i + max(get_amount(left, i - 1), get_amount(i + 1, right)) for i in range(left, right + 1))

        return get_amount(1, n)

print(Solution().getMoneyAmount(10))

'''

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.


Constraints:

1 <= n <= 200

'''


class Solution:
    def getMoneyAmount(self, n):
        need = [[0] * (n + 1) for _ in range(n + 1)]
        for lo in range(n, 0, -1):
            for hi in range(lo + 1, n + 1):
                need[lo][hi] = min(x + max(need[lo][x - 1], need[x + 1][hi])
                                   for x in range(lo, hi))
        return need[1][n]



    def getMoneyAmount3(self, n: int) -> int:
        import functools
        @functools.lru_cache(maxsize=None)
        def get_amount(left: int, right: int) -> int:
            if left >= right:
                return 0
            return min(i + max(get_amount(left, i - 1), get_amount(i + 1, right)) for i in range(left, right + 1))

        return get_amount(1, n)

print(Solution().getMoneyAmount(10))

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.   Constraints:  1 <= n <= 200