<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[0]), len(grid)
        
        rows = {}
        for j in range(m):
            rows[j] = [0] * (n+1)
            for i in range(n):
                rows[j][i+1] += rows[j][i] + grid[j][i]

        cols = {}
        for i in range(n):
            cols[i] = [0] * (m+1)
            for j in range(m):
                cols[i][j+1] += cols[i][j] + grid[j][i]

 
        def row_sm(i, j, k):
            return rows[j][i+k] - rows[j][i]
            #return sum(grid[j][i:i+k])
        
        def col_sm(i, j, k):
            return cols[i][j+k] - cols[i][j]
            #return sum(grid[y][i] for y in range(j, j+k))
        
        def diag1_sm(i, j, k):
            return sum(grid[j+x][i+x] for x in range(k))
        
        def diag2_sm(i, j, k):
            return sum(grid[j+x][i+k-x-1] for x in range(k))
        
        def is_magic(i, j, k):
            diag1 = diag1_sm(i, j, k)
            if diag2_sm(i, j, k) != diag1:
                return False
            if not all(col_sm(x, j, k) == diag1 for x in range(i, i+k)):
                return False
            if not all(row_sm(i, y, k) == diag1 for y in range(j, j+k)):
                return False
            return True
        
        for k in reversed(range(1, min(n, m) + 1)):
            for i in range(n-k+1):
                for j in range(m-k+1):
                    if is_magic(i, j, k):
                        return k
        return 1
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