<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 isMagicRow(self, start_row, end_row, start_column, end_column, grid):
        row_sum = -1
        for i in range(start_row, end_row+1):
            r_sum = 0
            for j in range(start_column, end_column+1):
                r_sum+=grid[i][j]
            if row_sum == -1:
                row_sum = r_sum
            if row_sum!= r_sum:
                return [False]
        return [True, row_sum]

    def isMagicColumn(self, start_row, end_row, start_column, end_column, grid):
        column_sum = -1
        for i in range(start_column, end_column+1):
            c_sum = 0
            for j in range(start_row, end_row+1):
                c_sum+= grid[j][i]
            if column_sum == -1:
                column_sum = c_sum
            if column_sum != c_sum:
                return [False]
        return [True, column_sum]
    
    def isMagicDiag(self, start_row, end_row, start_column, end_column, grid):
        left, right = 0,0
        sr = start_row
        for i in range(start_column, end_column+1):
            left+=grid[sr][i]
            sr+=1
        for i in range(end_column, start_column-1, -1):
            right+= grid[start_row][i]
            start_row+=1
        return [(left == right), left]

    def largestMagicSquare(self, grid: List[List[int]]) -> int:
        n = len(grid)
        m = len(grid[0])
        print(n,m)
        ans = 1
        for i in range(n):
            for j in range(m):
                incr = 1
                
                while i+incr < n and j+incr < m:
                    if self.isMagicRow(i, i+incr, j, j+incr, grid)[0] and self.isMagicColumn(i, i+incr, j, j+incr, grid)[0] and self.isMagicDiag(i, i+incr, j, j+incr, grid)[0]:
                        if self.isMagicRow(i, i+incr, j, j+incr, grid)[1] == self.isMagicColumn(i, i+incr, j, j+incr, grid)[1] == self.isMagicDiag(i, i+incr, j, j+incr, grid)[1]:
                            ans = max(ans, incr+1)

                    incr+=1
        return ans


3968