編寫一個程式，通過填充空格來解決數獨問題。

數獨的解法需 遵循如下規則：

數字1-9在每一行只能出現一次。
數字1-9在每一列只能出現一次。
數字1-9在每一個以粗實線分隔的3x3宮內只能出現一次。（請參考示例圖）
數獨部分空格內已填入了數字，空白格用'.'表示。

示例：

輸入：board = [["5","3",".",".","7",".",".",".","."],["6",".",".","1","9","5",".",".","."],[".","9","8",".",".",".",".","6","."],["8",".",".",".","6",".",".",".","3"],["4",".",".","8",".","3",".",".","1"],["7",".",".",".","2",".",".",".","6"],[".","6",".",".",".",".","2","8","."],[".",".",".","4","1","9",".",".","5"],[".",".",".",".","8",".",".","7","9"]]
輸出：[["5","3","4","6","7","8","9","1","2"],["6","7","2","1","9","5","3","4","8"],["1","9","8","3","4","2","5","6","7"],["8","5","9","7","6","1","4","2","3"],["4","2","6","8","5","3","7","9","1"],["7","1","3","9","2","4","8","5","6"],["9","6","1","5","3","7","2","8","4"],["2","8","7","4","1","9","6","3","5"],["3","4","5","2","8","6","1","7","9"]]

解釋：輸入的數獨如上圖所示，唯一有效的解決方案如下所示：

提示：

board.length == 9
board[i].length == 9
board[i][j] 是一位數字或者 '.'
題目資料 保證 輸入數獨僅有一個解

#include<iostream>
#include<stack>
#include<algorithm>
#include<string>
#include<vector>
using namespace std;
/*
對於每個需要填數字的格子帶入1到9，每代入一個數字都判定其是否合法，
如果合法就繼續下一次遞迴，結束時把數字設回 '.'，
判斷新加入的數字是否合法時，只需要判定當前數字是否合法，
不需要判定這個陣列是否為數獨陣列，因為之前加進的數字都是合法的，
這樣可以使程式更加高效一些，整體思路是這樣的，
但是實現起來可以有不同的形式。
一種實現形式是遞迴帶上橫縱座標，由於是一行一行的填數字，
且是從0行開始的，所以當i到達9的時候，
說明所有的數字都成功的填入了，直接返回 ture。
當j大於等於9時，當前行填完了，需要換到下一行繼續填，則繼續呼叫遞迴函式，
橫座標帶入 i+1。否則看若當前數字不為點，
說明當前位置不需要填數字，則對右邊的位置呼叫遞迴。
若當前位置需要填數字，則應該嘗試填入1到9內的所有數字，
讓c從1遍歷到9，每當試著填入一個數字，
都需要檢驗是否有衝突，使用另一個子函式 isValid 來檢驗是否合法，
假如不合法，則跳過當前數字。若合法，則將當前位置賦值為這個數字，
並對右邊位置呼叫遞迴，若遞迴函式返回 true，
則說明可以成功填充，直接返回 true。不行的話，需要重置狀態，
將當前位置恢復為點。若所有數字都嘗試了，還是不行，則最終返回 false。
*/

class Solution {
public:
	void solveSudoku(vector<vector<char>>& board) 
	{
		helper(board);
	}
	bool helper(vector<vector<char>>& board) 
	{
		for (int i = 0; i < 9; ++i) 
		{
			for (int j = 0; j < 9; ++j) 
			{
				if (board[i][j] != '.')
				{
					continue;
				}
				for (char c = '1'; c <= '9'; ++c) 
				{
					if (!isValid(board, i, j, c))
					{
						continue;
					}
					board[i][j] = c;
					if (helper(board))
					{
						return true;
					}
					board[i][j] = '.';
				}
				return false;
			}
		}
		return true;
	}
	bool isValid(vector<vector<char>>& board, int i, int j, char val) 
	{
		for (int k = 0; k < 9; ++k) 
		{
			if (board[k][j] != '.' && board[k][j] == val)
			{
				return false;
			}
			if (board[i][k] != '.' && board[i][k] == val)
			{
				return false;
			}
			int row = i / 3 * 3 + k / 3, col = j / 3 * 3 + k % 3;
			if (board[row][col] != '.' && board[row][col] == val)
			{
				return false;
			}
		}
		return true;
	}
};