<p>A <code>k x k</code> <strong>magic square</strong> is a <code>k x k</code> grid filled with integers such that every row sum, every column sum, and both diagonal sums are <strong>all equal</strong>. The integers in the magic square <strong>do not have to be distinct</strong>. Every <code>1 x 1</code> grid is trivially a <strong>magic square</strong>.</p>

<p>Given an <code>m x n</code> integer <code>grid</code>, return <em>the <strong>size</strong> (i.e., the side length </em><code>k</code><em>) of the <strong>largest magic square</strong> that can be found within this grid</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/05/29/magicsquare-grid.jpg" style="width: 413px; height: 335px;" />
<pre>
<strong>Input:</strong> grid = [[7,1,4,5,6],[2,5,1,6,4],[1,5,4,3,2],[1,2,7,3,4]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The largest magic square has a size of 3.
Every row sum, column sum, and diagonal sum of this magic square is equal to 12.
- Row sums: 5+1+6 = 5+4+3 = 2+7+3 = 12
- Column sums: 5+5+2 = 1+4+7 = 6+3+3 = 12
- Diagonal sums: 5+4+3 = 6+4+2 = 12
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/05/29/magicsquare2-grid.jpg" style="width: 333px; height: 255px;" />
<pre>
<strong>Input:</strong> grid = [[5,1,3,1],[9,3,3,1],[1,3,3,8]]
<strong>Output:</strong> 2
</pre>

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

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 50</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution:
    def largestMagicSquare(self, grid: List[List[int]]) -> int:
        rows, cols, result = len(grid), len(grid[0]), 0

        rowPrefix = [[0] * cols for _ in range(rows)]
        colPrefix = [[0] * cols for _ in range(rows)]

        for row in range(rows):
            for col in range(cols):
                rowPrefix[row][col] = grid[row][col] + (rowPrefix[row - 1][col] if row > 0 else 0)
                colPrefix[row][col] = grid[row][col] + (colPrefix[row][col - 1] if col > 0 else 0)

        for r in range(rows):
            for c in range(cols):
                dim = min(rows - r, cols - c)
                result = max(result, self.magicSquare(r, c, dim, rowPrefix, colPrefix, grid))
        
        return result
    
    def magicSquare(self, r, c, dim, rowPrefix, colPrefix, grid):
        
        for dim in range(dim, 0, -1):
            totalSum = colPrefix[r][c + dim - 1] - (colPrefix[r][c - 1] if c > 0 else 0)
            checks = left = right = 0

            for idx in range(dim):
                if totalSum != colPrefix[r + idx][c + dim - 1] - (colPrefix[r + idx][c - 1] if c > 0 else 0):
                    break
                if totalSum != rowPrefix[r + dim - 1][c + idx] - (rowPrefix[r - 1][c + idx] if r > 0 else 0):
                    break
                
                left += grid[r + idx][c + idx]
                right += grid[r + idx][c + dim - idx - 1]

                checks += 2

            if checks == dim * 2 and left == totalSum and right == totalSum:
                return dim

        return 0
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