In this section, we will try to solve the famous number maze problem called Sudoku. Sudoku is a 9 x 9 number grid, and the whole grid are also divided into 3 x 3 boxes There are some rules to solve the Sudoku.
We have to use digits 1 to 9 for solving this problem.
One digit cannot be repeated in one row, one column or in one 3 x 3 box.
Using the backtracking algorithm, we will try to solve the Sudoku problem. When some cell is filled with a digit, it checks whether it is valid or not. When it is not valid, it checks for other numbers. If all numbers are checked from 1-9, and no valid digit found to place, it backtracks to the previous option.
Input and Output
Input: This will take a 9 x 9 matrix as Sudoku grid. Some values are placed in the grid. The blank spaces are denoted by 0.Output: The final matrix (Sudoku grid) filled with numbers. If the solution does not exist, it will return false. 3 1 6 | 5 7 8 | 4 9 2 5 2 9 | 1 3 4 | 7 6 8 4 8 7 | 6 2 9 | 5 3 1 ------------------------ 2 6 3 | 4 1 5 | 9 8 7 9 7 4 | 8 6 3 | 1 2 5 8 5 1 | 7 9 2 | 6 4 3 ------------------------ 1 3 8 | 9 4 7 | 2 5 6 6 9 2 | 3 5 1 | 8 7 4 7 4 5 | 2 8 6 | 3 1 9
Algorithm
isPresentInCol(col, num)
Input: The column, and the targeted number.
Output − True when the number is present in the given column.
Begin for each row r in the grid, do if grid[r, col] = num, then return true done return false otherwise End
isPresentInRow(row, num)
Input − The row, and the targeted number.
Output − True when the number is present in the given column.
Begin for each column c in the grid, do if grid[row, c] = num, then return true done return false otherwise End
isPresentInBox(boxStartRow, boxStartCol, num)
Input − The starting row and column of a 3 x 3 box, and the targeted number.
Output − True when the number is present in the box.
Begin for each row r in boxStartRow to next 3 rows, do for each col r in boxStartCol to next 3 columns, do if grid[r, c] = num, then return true done done return false otherwise End
findEmptyPlace(row, col)
Input: row and column in the grid.
Output − If the grid[row, col] is empty, then return true, otherwise false.
Begin for each row r in the grid, do for each column c in the grid, do if grid[r, c] = 0, then return true done done return false End
isValidPlace(row, col, num)
Input: Row, a column of the grid, and number to check.
Output: True, when placing the number at position grid[row, col] is valid.
Begin if isPresentInRow(row, num) and isPresentInCol(col, num) and isPresntInBox(row – row mod 3, col – col mod 3, num) all are false, then return true End
solveSudoku(Sudoku Grid)
Input: The unsolved grid of Sudoku.
Output: Grid after solve.
Begin if no place in the grid is empty, then return true for number 1 to 9, do if isValidPlace(row, col, number), then grid[row, col] := number if solveSudoku = true, then return true grid[row, col] := 0 done return false End
Example
#include <iostream>
#define N 9
using namespace std;
int grid[N][N] = {
{3, 0, 6, 5, 0, 8, 4, 0, 0},
{5, 2, 0, 0, 0, 0, 0, 0, 0},
{0, 8, 7, 0, 0, 0, 0, 3, 1},
{0, 0, 3, 0, 1, 0, 0, 8, 0},
{9, 0, 0, 8, 6, 3, 0, 0, 5},
{0, 5, 0, 0, 9, 0, 6, 0, 0},
{1, 3, 0, 0, 0, 0, 2, 5, 0},
{0, 0, 0, 0, 0, 0, 0, 7, 4},
{0, 0, 5, 2, 0, 6, 3, 0, 0}
};
bool isPresentInCol(int col, int num) { //check whether num is present in col or not
for (int row = 0; row < N; row++)
if (grid[row][col] == num)
return true;
return false;
}
bool isPresentInRow(int row, int num) { //check whether num is present in row or not
for (int col = 0; col < N; col++)
if (grid[row][col] == num)
return true;
return false;
}
bool isPresentInBox(int boxStartRow, int boxStartCol, int num) { //check whether num is present in 3x3 box or not
for (int row = 0; row < 3; row++)
for (int col = 0; col < 3; col++)
if (grid[row+boxStartRow][col+boxStartCol] == num)
return true;
return false;
}
void sudokuGrid() { //print the sudoku grid after solve
for (int row = 0; row < N; row++) {
for (int col = 0; col < N; col++) {
if(col == 3 || col == 6)
cout << " | ";
cout << grid[row][col] <<" ";
}
if(row == 2 || row == 5) {
cout << endl;
for(int i = 0; i<N; i++)
cout << "---";
}
cout << endl;
}
}
bool findEmptyPlace(int &row, int &col) { //get empty location and update row and column
for (row = 0; row < N; row++)
for (col = 0; col < N; col++)
if (grid[row][col] == 0) //marked with 0 is empty
return true;
return false;
}
bool isValidPlace(int row, int col, int num) {
//when item not found in col, row and current 3x3 box
return !isPresentInRow(row, num) && !isPresentInCol(col, num) && !isPresentInBox(row - row%3 , col - col%3, num);
}
bool solveSudoku() {
int row, col;
if (!findEmptyPlace(row, col))
return true; //when all places are filled
for (int num = 1; num <= 9; num++) { //valid numbers are 1 - 9
if (isValidPlace(row, col, num)) { //check validation, if yes, put the number in the grid
grid[row][col] = num;
if (solveSudoku()) //recursively go for other rooms in the grid
return true;
grid[row][col] = 0; //turn to unassigned space when conditions are not satisfied
}
}
return false;
}
int main() {
if (solveSudoku() == true)
sudokuGrid();
else
cout << "No solution exists";
}Output
3 1 6 | 5 7 8 | 4 9 2 5 2 9 | 1 3 4 | 7 6 8 4 8 7 | 6 2 9 | 5 3 1 ------------------------ 2 6 3 | 4 1 5 | 9 8 7 9 7 4 | 8 6 3 | 1 2 5 8 5 1 | 7 9 2 | 6 4 3 ------------------------ 1 3 8 | 9 4 7 | 2 5 6 6 9 2 | 3 5 1 | 8 7 4 7 4 5 | 2 8 6 | 3 1 9