feat(csharp/backtracking): add csharp code in n queens (#485)

* feat(csharp/backtracking): add csharp code in n queens

* fix format

* Update n_queens.cs

---------

Co-authored-by: Yudong Jin <krahets@163.com>

codes/csharp/chapter_backtracking/n_queens.cs

@@ -0,0 +1,79 @@
+/**
+ * File: n_queens.cs
+ * Created Time: 2023-05-04
+ * Author: hpstory (hpstory1024@163.com)
+ */
+
+using hello_algo.utils;
+using NUnit.Framework;
+
+namespace hello_algo.chapter_backtracking;
+
+public class n_queens {
+    /* 回溯算法：N 皇后 */
+    static void backtrack(int row, int n, List<List<string>> state, List<List<List<string>>> res,
+            bool[] cols, bool[] diags1, bool[] diags2) {
+        // 当放置完所有行时，记录解
+        if (row == n) {
+            List<List<string>> copyState = new List<List<string>>();
+            foreach (List<string> sRow in state) {
+                copyState.Add(new List<string>(sRow));
+            }
+            res.Add(copyState);
+            return;
+        }
+        // 遍历所有列
+        for (int col = 0; col < n; col++) {
+            // 计算该格子对应的主对角线和副对角线
+            int diag1 = row - col + n - 1;
+            int diag2 = row + col;
+            // 剪枝：不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后
+            if (!(cols[col] || diags1[diag1] || diags2[diag2])) {
+                // 尝试：将皇后放置在该格子
+                state[row][col] = "Q";
+                cols[col] = diags1[diag1] = diags2[diag2] = true;
+                // 放置下一行
+                backtrack(row + 1, n, state, res, cols, diags1, diags2);
+                // 回退：将该格子恢复为空位
+                state[row][col] = "#";
+                cols[col] = diags1[diag1] = diags2[diag2] = false;
+            }
+        }
+    }
+
+    /* 求解 N 皇后 */
+    static List<List<List<string>>> nQueens(int n) {
+        // 初始化 n*n 大小的棋盘，其中 'Q' 代表皇后，'#' 代表空位
+        List<List<string>> state = new List<List<string>>();
+        for (int i = 0; i < n; i++) {
+            List<string> row = new List<string>();
+            for (int j = 0; j < n; j++) {
+                row.Add("#");
+            }
+            state.Add(row);
+        }
+        bool[] cols = new bool[n]; // 记录列是否有皇后
+        bool[] diags1 = new bool[2 * n - 1]; // 记录主对角线是否有皇后
+        bool[] diags2 = new bool[2 * n - 1]; // 记录副对角线是否有皇后
+        List<List<List<string>>> res = new List<List<List<string>>>();
+
+        backtrack(0, n, state, res, cols, diags1, diags2);
+
+        return res;
+    }
+
+    [Test]
+    public void Test() {
+        int n = 4;
+        List<List<List<string>>> res = nQueens(n);
+
+        Console.WriteLine("输入棋盘长宽为 " + n);
+        Console.WriteLine("皇后放置方案共有 " + res.Count + " 种");
+        foreach (List<List<string>> state in res) {
+            Console.WriteLine("--------------------");
+            foreach (List<string> row in state) {
+                PrintUtil.PrintList(row);
+            }
+        }
+    }
+}