<p>The <strong>n-queens</strong> puzzle is the problem of placing <code>n</code> queens on an <code>n x n</code> chessboard such that no two queens attack each other.</p>

<p>Given an integer <code>n</code>, return <em>the number of distinct solutions to the&nbsp;<strong>n-queens puzzle</strong></em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/13/queens.jpg" style="width: 600px; height: 268px;" />
<pre>
<strong>Input:</strong> n = 4
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are two distinct solutions to the 4-queens puzzle as shown.
</pre>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 9</code></li>
</ul>

class Solution {
public:
    int cnt = 0;
    bool is_valid(int r, int c, vector<string> & board) {
        // up, up-left and up-right
        vector<pair<int, int>> dirs = {{-1, 0}, {-1, -1}, {-1, 1}};
        for (auto & dir : dirs) {
            int curr = r, curc = c;
            while (0 <= curr and curr < board.size() and 0 <= curc and curc < board.size()) {
                if (board[curr][curc] == 'Q') return false;
                curr += dir.first;
                curc += dir.second;
            }
        }
        return true;
    }

    void backtrack(int n, int r, vector<string> & board) {
        if (r == n) {
            ++cnt;
            return;
        }
        for (int i = 0; i < n; ++i) {
            // If (r, i) doesn't intersect, place it. Recurse, then remove again to try another position
            if (is_valid(r, i, board)) {
                board[r][i] = 'Q';
                backtrack(n, r + 1, board);
                board[r][i] = '.';
            }
        }
    }
    
    int totalNQueens(int n) {
        string ref = "";
        for (int i = 0; i < n; ++i) {
            ref += '.';
        }
        vector<string> board(n, ref);
        backtrack(n, 0, board);
        return cnt;
    }
};
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