<p>You are given a 2D matrix <code>grid</code> of size <code>n x n</code>. Initially, all cells of the grid are colored white. In one operation, you can select any cell of indices <code>(i, j)</code>, and color black all the cells of the <code>j<sup>th</sup></code> column starting from the top row down to the <code>i<sup>th</sup></code> row.</p>

<p>The grid score is the sum of all <code>grid[i][j]</code> such that cell <code>(i, j)</code> is white and it has a horizontally adjacent black cell.</p>

<p>Return the <strong>maximum</strong> score that can be achieved after some number of operations.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[0,0,0,0,0],[0,0,3,0,0],[0,1,0,0,0],[5,0,0,3,0],[0,0,0,0,2]]</span></p>

<p><strong>Output:</strong> <span class="example-io">11</span></p>

<p><strong>Explanation:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2024/05/11/one.png" style="width: 300px; height: 200px;" />
<p>In the first operation, we color all cells in column 1 down to row 3, and in the second operation, we color all cells in column 4 down to the last row. The score of the resulting grid is <code>grid[3][0] + grid[1][2] + grid[3][3]</code> which is equal to 11.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[10,9,0,0,15],[7,1,0,8,0],[5,20,0,11,0],[0,0,0,1,2],[8,12,1,10,3]]</span></p>

<p><strong>Output:</strong> <span class="example-io">94</span></p>

<p><strong>Explanation:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2024/05/11/two-1.png" style="width: 300px; height: 200px;" />
<p>We perform operations on 1, 2, and 3 down to rows 1, 4, and 0, respectively. The score of the resulting grid is <code>grid[0][0] + grid[1][0] + grid[2][1] + grid[4][1] + grid[1][3] + grid[2][3] + grid[3][3] + grid[4][3] + grid[0][4]</code> which is equal to 94.</p>
</div>

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

<ul>
	<li><code>1 &lt;=&nbsp;n == grid.length &lt;= 100</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>0 &lt;= grid[i][j] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumScore(self, grid: List[List[int]]) -> int:
        state = [[0 for _ in range(i+1)] for i in range(len(grid)+1)]
        maxRow = [0]*(len(grid)+1)
        maxCol = [0]*(len(grid)+1)
        maxGrid = [[0 for _ in range(i+1)] for i in range(len(grid)+1)]
        sums = [[0 for _ in range(len(grid)-i)] for i in range(len(grid))]
        sumsN = [[0 for _ in range(len(grid)-i)] for i in range(len(grid))]
        for i in range(len(grid)):
            sums[i][0] = grid[i][0]
            for k in range(1,len(grid)-i):
                sums[i][k] = sums[i][k-1] + grid[i+k][0]
        for j in range(1,len(grid)):
            for i in range(len(grid)):
                sumsN[i][0] = grid[i][j]
                for k in range(1,len(grid)-i):
                    sumsN[i][k] = sumsN[i][k-1] + grid[i+k][j]
            for i in range(len(grid)+1):
                maxRow[i] = max(state[i])
                maxCol[i] = max(state[k][i] for k in range(i,len(grid)+1))
            maxGrid[len(grid)][0] = state[len(grid)][0]
            for k in range(1,len(grid)+1):
                maxGrid[len(grid)][k] = max(state[len(grid)][k],maxGrid[len(grid)][k-1])
                maxGrid[len(grid)-k][0] = max(state[len(grid)-k][0],maxGrid[len(grid)-k+1][0])
            for k in range(1,len(grid)):
                for i in range(len(grid)-1,k-1,-1):
                    maxGrid[i][k] = max(state[i][k],maxGrid[i+1][k],maxGrid[i][k-1])
            for i in range(1,len(grid)+1):
                for k in range(0,i):
                    state[i][k] = maxCol[i] + sumsN[k][i-k-1]
            for i in range(len(grid)+1):
                state[i][i] = maxGrid[i][i]
                for k in range(i-1,-1,-1):
                    state[i][i] = max(state[i][i],maxRow[k]+sums[k][i-k-1])
            sums, sumsN = sumsN, sums
        return max(max(x) for x in state)
18300