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

        def existMagicSquare(grid, m, n, side):
            maxRow, maxColumn = m - side, n - side

            for i in range(maxRow + 1):
                for j in range(maxColumn + 1):
                    if isMagicSquare(grid, i, j, side):
                        return True
            return False

        def isMagicSquare(grid, startRow, startColumn, side):
            rowSums = [0] * side
            columnSums = [0] * side
            diagonal1 = 0
            diagonal2 = 0

            for i in range(side):
                for j in range(side):
                    value = grid[startRow + i][startColumn + j]
                    rowSums[i] += value
                    columnSums[j] += value
                    if i == j:
                        diagonal1 += value
                    if i + j == side - 1:
                        diagonal2 += value

            magic_sum = rowSums[0]
            for r_sum in rowSums:
                if r_sum != magic_sum:
                    return False
            for c_sum in columnSums:
                if c_sum != magic_sum:
                    return False
            if diagonal1 != magic_sum or diagonal2 != magic_sum:
                return False
            
            return True

        for side in range(min(m, n), 1, -1):
            if existMagicSquare(grid, m, n, side):
                return side

        return 1        
16700