<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 isMagic(sq):
            n = len(sq)
            dist = {}
            sums = {}
            d1_sum = 0
            d2_sum = 0
            for i in range(n):
                c_sum = 0
                for j in range(n):
                    c_sum += sq[j][i]
                d1_sum += sq[i][i]
                d2_sum += sq[i][(n-1)-i]
                r_sum = sum(sq[i])
                if r_sum not in sums:
                    sums[r_sum] = r_sum
                if c_sum not in sums:
                    sums[c_sum] = c_sum
            if d1_sum not in sums:
                sums[d1_sum] = d1_sum
            if d2_sum not in sums:
                sums[d2_sum] = d2_sum
            if len(sums) > 1:
                return False
            return True

        out = 1
        m,n = len(grid),len(grid[0])
        sz = min([m,n])
        if sz < 2:
            return out
        for i in range(1,sz):
            count = 0
            for j in range(m-i):
                for k in range(n-i):
                    sq = [grid[j][k:k+i+1]]
                    for l in range(1,i+1):
                        sq.append(grid[j+l][k:k+i+1])
                    if isMagic(sq):
                        count += 1
            if count > 0:
                out = i+1
        return out
                        

            
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