Problem: Fill a 9x9 grid so that each row, column, and 3x3 subgrid contains all digits from 1 to 9.

Approach:

1. Use a recursive function to fill empty cells.
2. For each empty cell, try all possible numbers.
3. If placing a number leads to a valid solution, continue. If not, backtrack.

Code Example:

#include <iostream>
#include <vector>

using namespace std;

bool isValid(vector<vector<char>>& board, int row, int col, char num) {
    for (int i = 0; i < 9; i++) {
        if (board[row][i] == num || board[i][col] == num ||
            board[row - row % 3 + i / 3][col - col % 3 + i % 3] == num)
            return false;
    }
    return true;
}

bool solveSudoku(vector<vector<char>>& board) {
    for (int row = 0; row < 9; row++) {
        for (int col = 0; col < 9; col++) {
            if (board[row][col] == '.') {
                for (char num = '1'; num <= '9'; num++) {
                    if (isValid(board, row, col, num)) {
                        board[row][col] = num;
                        if (solveSudoku(board))
                            return true;
                        board[row][col] = '.';
                    }
                }
                return false;
            }
        }
    }
    return true;
}

void printBoard(vector<vector<char>>& board) {
    for (auto& row : board) {
        for (auto& cell : row) {
            cout << cell << " ";
        }
        cout << endl;
    }
}

int main() {
    vector<vector<char>> 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'}
    };
    solveSudoku(board);
    printBoard(board);
    return 0;
}