<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:
        def is_magic(matrix):
            n = len(matrix)
            first = sum(matrix[0])
            for row in matrix:
                if sum(row) != first:
                    return False
            first = sum([matrix[i][0] for i in range(n)])
            for j in range(1,n):
                total = sum([matrix[i][j] for i in range(n)])
                if total != first:
                    return False
            d1 = sum([matrix[i][i] for i in range(n)])
            d2 = sum([matrix[i][n-i-1] for i in range(n)])
            return d1 == d2 == first

        size = 1
        m , n = len(grid) , len(grid[0])
        for i in range(m):
            for j in range(n):
                x , y = i+1 , j+1
                row_sum = col_sum = grid[i][j]
                while x < m and y < n:
                    row_sum += grid[i][y]
                    col_sum += grid[x][j]
                    if row_sum == col_sum:
                        # print(f'x = {x} and y = {y}')
                        subgrid = [row[j:y+1] for row in grid[i:x+1]]
                        if is_magic(subgrid):
                            # print(f'Magic , x = {x} and i = {i}')
                            size = max(x-i + 1,size)
                    x += 1
                    y += 1
        return size
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