<p>A <code>3 x 3</code> <strong>magic square</strong> is a <code>3 x 3</code> grid filled with distinct numbers <strong>from </strong>1<strong> to </strong>9 such that each row, column, and both diagonals all have the same sum.</p>

<p>Given a <code>row x col</code> <code>grid</code> of integers, how many <code>3 x 3</code> magic square subgrids are there?</p>

<p>Note: while a magic square can only contain numbers from 1 to 9, <code>grid</code> may contain numbers up to 15.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/11/magic_main.jpg" style="width: 322px; height: 242px;" />
<pre>
<strong>Input:</strong> grid = [[4,3,8,4],[9,5,1,9],[2,7,6,2]]
<strong>Output:</strong> 1
<strong>Explanation: </strong>
The following subgrid is a 3 x 3 magic square:
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/11/magic_valid.jpg" style="width: 242px; height: 242px;" />
while this one is not:
<img alt="" src="https://assets.leetcode.com/uploads/2020/09/11/magic_invalid.jpg" style="width: 242px; height: 242px;" />
In total, there is only one magic square inside the given grid.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [[8]]
<strong>Output:</strong> 0
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>row == grid.length</code></li>
	<li><code>col == grid[i].length</code></li>
	<li><code>1 &lt;= row, col &lt;= 10</code></li>
	<li><code>0 &lt;= grid[i][j] &lt;= 15</code></li>
</ul>

class Solution {
public:
    
    bool solve(int r, int c, vector<vector<int>>& arr){
        set<int> sum, no; 
        int diag = 0; 
        for(int i = r; i<r+3; i++){
            int rowsum = 0; 
            for(int j = c; j<c+3; ++j){
                rowsum += arr[i][j]; 
                if ( arr[i][j] < 1 || arr[i][j] > 9) return false; 
                no.insert(arr[i][j]); 
            }
            sum.insert(rowsum); 
        }
        diag = 0; 

        int i = r, j = c+2; 
        while ( i < r + 3){
            diag += arr[i][j]; 
            i++, j--;
        }
        sum.insert(diag); 
        i = r, j = c, diag = 0; 
        while ( i < r + 3){
            diag += arr[i][j]; 
            i++, j++; 
        }
        sum.insert(diag); 

        if ( no.size() != 9 || sum.size() > 1) return false; 

        for(int j = c; j<c+3; ++j){
            int colsum = 0; 
            for(int i = r; i<r+3; ++i){
                colsum += arr[i][j]; 
            }
            sum.insert(colsum); 
        }

        return sum.size() == 1; 
    }

    int numMagicSquaresInside(vector<vector<int>>& grid) {
        int n = grid.size(); 
        int m = grid[0].size();

        int ans = 0; 
        for(int i = 0; i<n-2; ++i){
            for(int j = 0; j<m-2; ++j){
                if ( solve(i, j, grid) ) {
                    ans++; 
                }
                    
            }
        }
        return ans; 
    }
};
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