<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:
        R = len(board)
        C = len(board[0])
        MAX = 9999999999
        
        for r in range(R):
            oldrow = board[r]
            newboard = [(k,oldrow[k]) for k in range(len(oldrow))]
            newboard.sort(key=lambda k:-k[1])
            board[r] = newboard[:3]
        
        C = 3
        
        
        @cache
        def place(row, taken):
            if len(taken) == 3:
                #print(taken)
                return 0
            if row == R:
                return -MAX
            
            best = -MAX
            ntaken = list(taken)
            cand2 = place(row+1, taken)
            for take in range(C):
                if board[row][take][0] not in taken:
                    ntaken.append(board[row][take][0])
                    cand1 = place(row+1, tuple(sorted(ntaken))) + board[row][take][1]
                    ntaken.remove(board[row][take][0])
                    best = max(best, cand1)
                    
            return max(best, cand2)
    
        return place(0, tuple())
    '''
    [[95,-90,-62,-94],
    [62,59,-21,46],
    [23,24,19,9],
    [-79,-12,91,16]]
    
    [[95,-90,-62,-94],[62,59,-21,46],[23,24,19,9],[-79,-12,91,16]]
    '''
147428