## 375. Guess Number Higher or Lower II

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

1. I pick a number between `1` and `n`.
2. You guess a number.
3. If you guess the right number, you win the game.
4. 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.
5. 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`

## Rust Solution

``````struct Solution;
use std::collections::HashMap;

impl Solution {
fn get_money_amount(n: i32) -> i32 {
let mut memo: HashMap<(i32, i32), i32> = HashMap::new();
Self::dp(1, n, &mut memo)
}

fn dp(left: i32, right: i32, memo: &mut HashMap<(i32, i32), i32>) -> i32 {
if left == right {
0
} else {
if let Some(&res) = memo.get(&(left, right)) {
return res;
}
let mut res = std::i32::MAX;
for i in left..right {
let a = if i != left {
Self::dp(left, i - 1, memo)
} else {
0
};
let b = if i != right {
Self::dp(i + 1, right, memo)
} else {
0
};
res = res.min(i + a.max(b));
}
memo.insert((left, right), res);
res
}
}
}

#[test]
fn test() {
let n = 10;
let res = 16;
assert_eq!(Solution::get_money_amount(n), res);
}
``````

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