<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>all distinct solutions to the <strong>n-queens puzzle</strong></em>. You may return the answer in <strong>any order</strong>.</p>

<p>Each solution contains a distinct board configuration of the n-queens&#39; placement, where <code>&#39;Q&#39;</code> and <code>&#39;.&#39;</code> both indicate a queen and an empty space, respectively.</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> [[&quot;.Q..&quot;,&quot;...Q&quot;,&quot;Q...&quot;,&quot;..Q.&quot;],[&quot;..Q.&quot;,&quot;Q...&quot;,&quot;...Q&quot;,&quot;.Q..&quot;]]
<strong>Explanation:</strong> There exist two distinct solutions to the 4-queens puzzle as shown above
</pre>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> [[&quot;Q&quot;]]
</pre>

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

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

class Solution {
public:
    vector<vector<string>> solveNQueens(int n) {
        vector<vector<int>> validPositions(n, vector<int>(n));
        vector<string> currBoard;
        string boardString = "";
        for (int i = 0; i < n; i++) {
            boardString += ".";
        }

        for (int i = 0; i < n; i++) {
            currBoard.push_back(boardString);
        }

        calcBoard(0, n, validPositions, currBoard);
        return answers;
    }

private:
    void calcBoard(int col, int N, vector<vector<int>> validPositions, vector<string> currBoard) {
        if (col == N) {
            answers.push_back(currBoard);
            return;
        }

        for (int i = 0; i < N; i++) {
            if (validPositions[i][col] == EMPTY) {
                currBoard[i][col] = 'Q';
                addQueenLOS(validPositions, i, col);
                calcBoard(col + 1, N, validPositions, currBoard);
                
                currBoard[i][col] = '.';
                removeQueenLOS(validPositions, i, col);
            }
        }
    }

    void addQueenLOS(vector<vector<int>> &validPositions, int queenRow, int queenCol) {
        updateBoard(validPositions, queenRow, queenCol, 1);
    }

    void removeQueenLOS(vector<vector<int>> &validPositions, int queenRow, int queenCol) {
        updateBoard(validPositions, queenRow, queenCol, -1);
    }

    void updateBoard(vector<vector<int>> &validPositions, int queenRow, int queenCol, int val) {
        int N = validPositions.size();
        // queen position
        validPositions[queenRow][queenCol] = validPositions[queenRow][queenCol] + val;

        // down
        for (int i = queenRow + 1; i < N; i++) {
            validPositions[i][queenCol] = validPositions[i][queenCol] + val;
        }

        // up
        for (int i = queenRow - 1; i >= 0; i--) {
            validPositions[i][queenCol] = validPositions[i][queenCol] + val;
        }

        // left
        for (int j = queenCol - 1; j >= 0; j--) {
            validPositions[queenRow][j] = validPositions[queenRow][j] + val;
        }

        // right
        for (int j = queenCol + 1; j < N; j++) {
            validPositions[queenRow][j] = validPositions[queenRow][j] + val;
        }

        // down + right
        for (int i = queenRow + 1, j = queenCol + 1; i < N && j < N; i++, j++) {
            validPositions[i][j] = validPositions[i][j] + val;
        }

        // up + left
        for (int i = queenRow - 1, j = queenCol - 1; i >= 0 && j >= 0; i--, j--) {
            validPositions[i][j] = validPositions[i][j] + val;
        }

        // up + right
        for (int i = queenRow - 1, j = queenCol + 1; i >= 0 && j < N; i--, j++) {
            validPositions[i][j] = validPositions[i][j] + val;
        }

        // down + left
        for (int i = queenRow + 1, j = queenCol - 1; i < N && j >= 0; i++, j--) {
            validPositions[i][j] = validPositions[i][j] + val;
        }
    }

    vector<vector<string>> answers;

    const int EMPTY = 0;
};
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