<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>

import heapq
class Solution:
    def maximumValueSum(self, board: List[List[int]]) -> int:
        rows = ()  
        cache = {}
        
        max_vals = []
        for j in range(len(board[0])):
            heap = []
            heap_indexes = []
            for i in range(len(board)):
                heapq.heappush(heap, (-1*board[i][j], i))
            for i in range(3):
                val, index = heapq.heappop(heap)
                val=val*-1
                heap_indexes.append(index)
            max_vals.append(heap_indexes)

        def place(j, n):
            nonlocal rows
            if n<0:
                return float('-inf')
            if n==0:
                return 0
            if (j,n,rows) in cache:
                return cache[(j,n,rows)]
            
            if n<0 or j>=len(board[0]):
                return float("-inf")
            
            ans = float("-inf")
            

            ans = float('-inf')
            for i in range(3):
                index = max_vals[j][i]
                if index not in rows and n>0:
                    rows = rows +(index,)
                    ans = max(place(j+1, n-1)+board[index][j], ans) 
                    rows = rows[:-1]
            ans = max(ans, place(j+1, n))
            cache[(j,n,rows)] = ans 
            return ans 
                    
        return place(0, 3)
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