<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:
    static bool isMagic(int i, int j, vector<vector<int>>& grid){
    //    if (grid[i][j] != 5) return 0;

        bitset<10> once; // bitset of size 10 to track digits 1-9
        int rowSum[3] = {0}, colSum[3] = {0};

        for(int a=i-1; a <= i+1; a++) {
            for(int b=j-1; b <=j+1; b++) {
                int x= grid[a][b];
                if (x<1 || x>9) return 0; // invalid number for a magic square
                rowSum[a-i+1] += x;
                colSum[b-j+1] += x;
                once.flip(x);
            }
        }
        // numbers 1-9 are present exactly once
        if (once.count() != 9) return 0;

        // all rows and columns sum to 15
        if (!all_of(rowSum, rowSum+3, [](int sum){ return sum == 15; })) return 0;
        if (!all_of(colSum, colSum+3, [](int sum){ return sum == 15; })) return 0;

        // Check if diagonals sum to 15
        return (grid[i-1][j-1]+grid[i+1][j+1]==10 && grid[i+1][j-1]+grid[i-1][j+1]== 10);
    }

    static int numMagicSquaresInside(vector<vector<int>>& grid) {
        const int r = grid.size(), c = grid[0].size();
        if (r < 3 || c < 3) return 0;

        int cnt = 0;
        for(int i = 1; i < r-1; i++) {
            for(int j = 1; j < c-1; j++) {
                if (grid[i][j]==5) 
                    cnt+=isMagic(i, j, grid);
            }
        }
        return cnt;
    }
};


// auto init = []() {
//     ios::sync_with_stdio(false);
//     cin.tie(nullptr);
//     cout.tie(nullptr);
//     return 'c';
// }();
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