# Guess Number Higher Or Lower II Problem

## Description

LeetCode Problem 375.

We are playing the Guessing Game. The game will work as follows:

- I pick a number between1andn.
- 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 numberx, you will payxdollars. If you run out of money, you lose the game.

Given a particularn, returnthe minimum amount of money you need toguarantee a win regardless of what number I pick.

Example 1:

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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:

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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:

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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

## Sample C++ Code

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class Solution {
public:
int getMoneyAmount(int n) {
vector<vector<int>> dp(n+1, vector<int>(n+1, 0));
for(int len=2; len<=n; ++len){
for(int begin=0; begin<=n-len; ++begin){
int end = begin + len;
for(int i=begin; i<end; ++i){
int numPicked = i+1;
if(i == begin){
dp[begin][end] = numPicked + dp[begin+1][end];
} else {
dp[begin][end] = min(dp[begin][end], max(dp[begin][i], dp[i+1][end]) + numPicked);
}
}
}
}
return dp[0][n];
}
};