<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:
        '''
        given that a k x k where k==1 is a magic square, we will want to start with the biggest one we can fit,
        then explore all options of it, then reduce it by 1 if we didn't find anything.
        we will stop at k==2 if not found, and return 1.

        since the grid is n x m and not necessarily square, we will take the min(n,m) and use as our initial k.
        
        our main loop will go through from k to 2, scan all the grid from top-left to bottom-k,right-k, and call
        our isMagicSquare helper that evaluates a given top-left and k and return True if it is magical.
        '''

        def isMagicSquare(r:int, c:int, k:int) -> bool:
            sums=[0]*((2*k)+2)
            for i in range(k):
                for j in range(k):
                    # sum rows
                    sums[j]+=grid[r+i][c+j]
                    # sum cols
                    sums[k+j]+=grid[r+j][c+i]
                # sum left diag
                sums[-1]+=grid[r+i][c+i]
                # sum right diag
                sums[-2]+=grid[r+i][c+k-i-1]
            return all(x==sums[0] for x in sums)
        
        W=len(grid[0])
        H=len(grid)
        K=min(W,H)
        if K>1:
            for k in range(K,1,-1):
                for row in range(H+1-k):
                    for col in range(W+1-k):
                        if isMagicSquare(row,col,k): 
                            return k

        return 1
        
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