<p>We are playing the Guessing Game. The game will work as follows:</p>

<ol>
	<li>I pick a number between&nbsp;<code>1</code>&nbsp;and&nbsp;<code>n</code>.</li>
	<li>You guess a number.</li>
	<li>If you guess the right number, <strong>you win the game</strong>.</li>
	<li>If you guess the wrong number, then I will tell you whether the number I picked is <strong>higher or lower</strong>, and you will continue guessing.</li>
	<li>Every time you guess a wrong number&nbsp;<code>x</code>, you will pay&nbsp;<code>x</code>&nbsp;dollars. If you run out of money, <strong>you lose the game</strong>.</li>
</ol>

<p>Given a particular&nbsp;<code>n</code>, return&nbsp;<em>the minimum amount of money you need to&nbsp;<strong>guarantee a win regardless of what number I pick</strong></em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/10/graph.png" style="width: 505px; height: 388px;" />
<pre>
<strong>Input:</strong> n = 10
<strong>Output:</strong> 16
<strong>Explanation:</strong> The winning strategy is as follows:
- The range is [1,10]. Guess 7.
&nbsp;   - If this is my number, your total is $0. Otherwise, you pay $7.
&nbsp;   - If my number is higher, the range is [8,10]. Guess 9.
&nbsp;       - If this is my number, your total is $7. Otherwise, you pay $9.
&nbsp;       - If my number is higher, it must be 10. Guess 10. Your total is $7 + $9 = $16.
&nbsp;       - If my number is lower, it must be 8. Guess 8. Your total is $7 + $9 = $16.
&nbsp;   - If my number is lower, the range is [1,6]. Guess 3.
&nbsp;       - If this is my number, your total is $7. Otherwise, you pay $3.
&nbsp;       - If my number is higher, the range is [4,6]. Guess 5.
&nbsp;           - If this is my number, your total is $7 + $3 = $10. Otherwise, you pay $5.
&nbsp;           - If my number is higher, it must be 6. Guess 6. Your total is $7 + $3 + $5 = $15.
&nbsp;           - If my number is lower, it must be 4. Guess 4. Your total is $7 + $3 + $5 = $15.
&nbsp;       - If my number is lower, the range is [1,2]. Guess 1.
&nbsp;           - If this is my number, your total is $7 + $3 = $10. Otherwise, you pay $1.
&nbsp;           - 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.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 0
<strong>Explanation:</strong>&nbsp;There is only one possible number, so you can guess 1 and not have to pay anything.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 2
<strong>Output:</strong> 1
<strong>Explanation:</strong>&nbsp;There are two possible numbers, 1 and 2.
- Guess 1.
&nbsp;   - If this is my number, your total is $0. Otherwise, you pay $1.
&nbsp;   - If my number is higher, it must be 2. Guess 2. Your total is $1.
The worst case is that you pay $1.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 200</code></li>
</ul>

class Solution:
    def getMoneyAmount(self, n: int) -> int:
        dp = [ [float( "inf" )]*(n+1) for _ in range(n+1) ] # f(i, j) return the required money when guessing [i, j)
        # initialize
        for i in range(n+1):
            for j in range(i+1):
                dp[i][j] = 0

        for dist in range(1, n):
            for start in range(1, n+1):
                if start+dist > n:
                    break
                else:
                    end = start+dist
                    for pivot in range(start, end+1):
                        if pivot == end:
                            res = pivot + dp[start][pivot-1]
                        else:
                            res = pivot + max(dp[start][pivot-1], dp[pivot+1][end])
                        dp[start][end] = min(dp[start][end], res)
        print(dp)
        return dp[1][n]
17800