<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])
        ans = 1
        preSumRow = [[0] * (m + 1) for _ in range(n)]
        preSumCol = [[0] * (n + 1) for _ in range(m)]
        for r in range(n):
            for c in range(m):
                preSumRow[r][c + 1] = preSumRow[r][c] + grid[r][c]
                preSumCol[c][r + 1] = preSumCol[c][r] + grid[r][c]
        def getSumRow(row, l, r):  
            return preSumRow[row][r] - preSumRow[row][l]

        def getSumCol(col, l, r):  
            return preSumCol[col][r] - preSumCol[col][l]
        
        for i in range(n):
            for j in range(m):
                l,b = n-i,m-j
                s = min(l,b)
                while s>1:
                    diag,anti_diag = 0,0
                    row_sum = []
                    col_sum = []
                    flag = True
                    for k in range(s):
                        row_sum.append(getSumRow(i+k,j,j+s))
                        col_sum.append(getSumCol(j+k,i,i+s))
                        diag += grid[i+k][j+k]
                        anti_diag += grid[i + k][j+s-1-k]
                    if row_sum[0] == diag == anti_diag:
                        for r in range(s):
                            if row_sum[0] == row_sum[r] == col_sum[r]:
                                continue
                            else:
                                flag = False
                    else:
                        flag = False
                    if flag == True:
                        ans = max(ans,s)
                    s-=1
        return ans
                        

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