<p>You are given a <code>m x n</code> 2D array <code>board</code> representing a chessboard, where <code>board[i][j]</code> represents the <strong>value</strong> of the cell <code>(i, j)</code>.</p>

<p>Rooks in the <strong>same</strong> row or column <strong>attack</strong> each other. You need to place <em>three</em> rooks on the chessboard such that the rooks <strong>do not</strong> <strong>attack</strong> each other.</p>

<p>Return the <strong>maximum</strong> sum of the cell <strong>values</strong> on which the rooks are placed.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = </span>[[-3,1,1,1],[-3,1,-3,1],[-3,2,1,1]]</p>

<p><strong>Output:</strong> 4</p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/08/08/rooks2.png" style="width: 294px; height: 450px;" /></p>

<p>We can place the rooks in the cells <code>(0, 2)</code>, <code>(1, 3)</code>, and <code>(2, 1)</code> for a sum of <code>1 + 1 + 2 = 4</code>.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = [[1,2,3],[4,5,6],[7,8,9]]</span></p>

<p><strong>Output:</strong> <span class="example-io">15</span></p>

<p><strong>Explanation:</strong></p>

<p>We can place the rooks in the cells <code>(0, 0)</code>, <code>(1, 1)</code>, and <code>(2, 2)</code> for a sum of <code>1 + 5 + 9 = 15</code>.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = [[1,1,1],[1,1,1],[1,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>We can place the rooks in the cells <code>(0, 2)</code>, <code>(1, 1)</code>, and <code>(2, 0)</code> for a sum of <code>1 + 1 + 1 = 3</code>.</p>
</div>

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

<ul>
	<li><code>3 &lt;= m == board.length &lt;= 100</code></li>
	<li><code>3 &lt;= n == board[i].length &lt;= 100</code></li>
	<li><code>-10<sup>9</sup> &lt;= board[i][j] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumValueSum(self, board: List[List[int]]) -> int:
        m, n = len(board), len(board[0])

        arr = []

        for i in range(m):
            a = []
            for j in range(n):
                a.append((board[i][j],j))
            a.sort()
            arr.append([a[-1],a[-2],a[-3]])

        dp = [[[-float("inf") for _ in range(n)] for _ in range(n)] for _ in range(m)]

        for r in range(m):
            for r2 in range(r):
                for v, c in arr[r]:
                    for v2, c2 in arr[r2]:
                        if c == c2: continue
                        dp[r][c][c2] = max(dp[r][c][c2], v + v2)
            if r-1 >= 0:
                for c in range(n):
                    for c2 in range(n):
                        if c == c2: continue
                        dp[r][c][c2] = max(dp[r][c][c2], dp[r-1][c][c2])

        res = -float("inf")
        for r in range(1, m):  
            for v, c in arr[r]:
                for c2 in range(n):
                    if c2 == c: continue
                    for c3 in range(n):
                        if c2 == c3 or c == c3: continue
                        res = max(res, v + dp[r-1][c2][c3])

        return res


        """


        [-57,-56, 69,-75]
        [-72, 63,-96, 66]
        [ 46,-18, 37,-38]
        [ 85,-13,-33,-37]

        """
58361