<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:
        n, m = len(grid), len(grid[0])

        rowSums = [grid[i].copy() for i in range(n)]
        colSums = [grid[i].copy() for i in range(n)]

        for i in range(n):
            for j in range(1, m):
                rowSums[i][j] += rowSums[i][j - 1]
        
        for i in range(1, n):
            for j in range(m):
                colSums[i][j] += colSums[i - 1][j]
        
        def check(k):
            for r1 in range(n - k + 1):
                r2 = r1 + k
                for c1 in range(m - k + 1):
                    c2 = c1 + k
                    sums = set()

                    for i in range(r1, r2):
                        cur = rowSums[i][c2 - 1]
                        if c1 > 0:
                            cur -= rowSums[i][c1 - 1]
                        
                        sums.add(cur)
                    
                    for j in range(c1, c2):
                        cur = colSums[r2 - 1][j]
                        if r1 > 0:
                            cur -= colSums[r1 - 1][j]

                        sums.add(cur)
                    
                    cur = 0
                    for l in range(k):
                        r, c = r1 + l, c1 + l
                        cur += grid[r][c]
                    
                    sums.add(cur)

                    cur = 0
                    for l in range(k):
                        r, c = r1 + l, c2 - l - 1
                        cur += grid[r][c]
                    
                    sums.add(cur)
                    
                    if len(sums) == 1:
                        return True
            
            return False
        
        x = min(n, m) + 1
        maxK = -1
        for k in range(1, x):
            if check(k):
                maxK = max(maxK, k)  

        return maxK
1031