> For the complete documentation index, see [llms.txt](https://wentao-shao.gitbook.io/leetcode/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wentao-shao.gitbook.io/leetcode/graph-search/51.n-queens.md).

# 51.N-Queens

## 51. N Queens

## 题目地址

<https://leetcode.com/problems/n-queens/>

<https://www.lintcode.com/problem/n-queens/description>

## 题目描述

```
The n-queens puzzle is the problem of placing n queens on an n×n chessboard such that no two queens attack each other(Any two queens can't be in the same row, column, diagonal line).

Given an integer n, return all distinct solutions to the n-queens puzzle.

Each solution contains a distinct board configuration of the n-queens' placement, where 'Q' and '.' each indicate a queen and an empty space respectively.
```

## 代码

### Approach 1: Recursion

Time: `O(N!)` Space: `O(N)`

```java
class Solution {
    public List<List<String>> solveNQueens(int n) {
        List<List<String>> ans = new ArrayList();
        if (n <= 0) return ans;

        List<Integer> cols = new ArrayList();
        dfs(ans, cols, n);   

        return ans;
    }

    private void dfs(List<List<String>> ans, List<Integer> cols, int n) {
        if (cols.size() == n) {
            ans.add(output(cols));
            return;
        }


        for (int col = 0; col < n; col++) {
            if (isValid(cols, col)) {
                cols.add(col);
                dfs(ans, cols, n);
                cols.remove(cols.size() - 1);
            }
        }
    }

    private boolean isValid(List<Integer> cols, int col) {
        int row = cols.size();
        for (Integer r = 0; r < row; r++) {
            Integer c = cols.get(r);
            if (c == col) return false;
            if (row - col == r - c)  return false;
            if (row + col == r + c) return false;
        }
        return true;
    }

    private List<String> output(List<Integer> cols) {
        List<String> ans = new ArrayList();
        int n = cols.size();
        for (int r = 0; r < n; r++) {
            StringBuilder sb = new StringBuilder();
            for (int c = 0; c < n; c++) {
                if (c == cols.get(r)) {
                    sb.append("Q");
                } else {
                    sb.append(".");
                }
            }
            ans.add(sb.toString());
        }

        return ans;
    }
}
```