Program for Sudoku Generator

Background:
Following are the rules of Sudoku for a player. 

1. In all 9 sub matrices 3×3 the elements should be 1-9, without repetition.
2. In all rows there should be elements between 1-9 , without repetition.
3. In all columns there should be elements between 1-9 , without repetition.

The task is to generate a 9 x 9 Sudoku grid that is valid, i.e., a player can fill the grid following above set of rules.

A simple naïve solution can be. 

1. Randomly take any number 1-9.
2. Check if it is safe to put in the cell.(row , column and box)
3. If safe, place it and increment to next location and go to step 1.
4. If not safe, then without incrementing go to step 1.
5. Once matrix is fully filled, remove k no. of elements randomly to complete game.

Improved Solution : We can improve the solution, if we understand a pattern in this game. We can observe that all 3 x 3 matrices, which are diagonally present are independent of other 3 x 3 adjacent matrices initially, as others are empty. 

   3 8 5 0 0 0 0 0 0 
   9 2 1 0 0 0 0 0 0 
   6 4 7 0 0 0 0 0 0 
   0 0 0 1 2 3 0 0 0 
   0 0 0 7 8 4 0 0 0 
   0 0 0 6 9 5 0 0 0 
   0 0 0 0 0 0 8 7 3 
   0 0 0 0 0 0 9 6 2 
   0 0 0 0 0 0 1 4 5 

(We can observe that in above matrix, the diagonal matrices are independent of other empty matrices initially). So if we fill them first, then we will only have to do box check and thus column/row check not required.

Secondly, while we fill rest of the non-diagonal elements, we will not have to use random generator, but we can fill recursively by checking 1 to 9. 

Following is the improved logic for the problem.
1. Fill all the diagonal 3x3 matrices.
2. Fill recursively rest of the non-diagonal matrices.
   For every cell to be filled, we try all numbers until
   we find a safe number to be placed.  
3. Once matrix is fully filled, remove K elements
   randomly to complete game.

C++

/* C++ program for Sudoku generator  */
#include <bits/stdc++.h>
using namespace std;

class Sudoku {
public:
    int** mat;
    int N;

    // number of columns/rows.
    int SRN;

    // square root of N
    int K;
    // No. Of missing digits

    // Constructor
    Sudoku(int N, int K)
    {
        this->N = N;
        this->K = K;

        // Compute square root of N
        double SRNd = sqrt(N);
        SRN = (int)SRNd;
        mat = new int*[N];

        // Create a row for every pointer
        for (int i = 0; i < N; i++)
        {

            // Note : Rows may not be contiguous
            mat[i] = new int[N];

            // Initialize all entries as false to indicate
            // that there are no edges initially
            memset(mat[i], 0, N * sizeof(int));
        }
    }

    // Sudoku Generator
    void fillValues()
    {

        // Fill the diagonal of SRN x SRN matrices
        fillDiagonal();

        // Fill remaining blocks
        fillRemaining(0, SRN);

        // Remove Randomly K digits to make game
        removeKDigits();
    }

    // Fill the diagonal SRN number of SRN x SRN matrices
    void fillDiagonal()
    {
        for (int i = 0; i < N; i = i + SRN)
        {

            // for diagonal box, start coordinates->i==j
            fillBox(i, i);
        }
    }
    // Returns false if given 3 x 3 block contains num.
    bool unUsedInBox(int rowStart, int colStart, int num)
    {
        for (int i = 0; i < SRN; i++) {
            for (int j = 0; j < SRN; j++) {
                if (mat[rowStart + i][colStart + j]
                    == num) {
                    return false;
                }
            }
        }
        return true;
    }
    // Fill a 3 x 3 matrix.
    void fillBox(int row, int col)
    {
        int num;
        for (int i = 0; i < SRN; i++) {
            for (int j = 0; j < SRN; j++) {
                do {
                    num = randomGenerator(N);
                } while (!unUsedInBox(row, col, num));
                mat[row + i][col + j] = num;
            }
        }
    }
    // Random generator
    int randomGenerator(int num)
    {
        return (int)floor(
            (float)(rand() / double(RAND_MAX) * num + 1));
    }
    // Check if safe to put in cell
    bool CheckIfSafe(int i, int j, int num)
    {
        return (
            unUsedInRow(i, num) && unUsedInCol(j, num)
            && unUsedInBox(i - i % SRN, j - j % SRN, num));
    }
    // check in the row for existence
    bool unUsedInRow(int i, int num)
    {
        for (int j = 0; j < N; j++) {
            if (mat[i][j] == num) {
                return false;
            }
        }
        return true;
    }
    // check in the row for existence
    bool unUsedInCol(int j, int num)
    {
        for (int i = 0; i < N; i++) {
            if (mat[i][j] == num) {
                return false;
            }
        }
        return true;
    }
    // A recursive function to fill remaining
    // matrix
    bool fillRemaining(int i, int j)
    {
        // System.out.println(i+" "+j);
        if (j >= N && i < N - 1) {
            i = i + 1;
            j = 0;
        }
        if (i >= N && j >= N) {
            return true;
        }
        if (i < SRN) {
            if (j < SRN) {
                j = SRN;
            }
        }
        else if (i < N - SRN) {
            if (j == (int)(i / SRN) * SRN) {
                j = j + SRN;
            }
        }
        else {
            if (j == N - SRN) {
                i = i + 1;
                j = 0;
                if (i >= N) {
                    return true;
                }
            }
        }
        for (int num = 1; num <= N; num++) {
            if (CheckIfSafe(i, j, num)) {
                mat[i][j] = num;
                if (fillRemaining(i, j + 1)) {
                    return true;
                }
                mat[i][j] = 0;
            }
        }
        return false;
    }
    // Remove the K no. of digits to
    // complete game
    void removeKDigits()
    {
        int count = K;
        while (count != 0) {
            int cellId = randomGenerator(N * N) - 1;
            // System.out.println(cellId);
            // extract coordinates i and j
            int i = (cellId / N);
            int j = cellId % 9;
            if (j != 0) {
                j = j - 1;
            }
            // System.out.println(i+" "+j);
            if (mat[i][j] != 0) {
                count--;
                mat[i][j] = 0;
            }
        }
    }
    // Print sudoku
    void printSudoku()
    {
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < N; j++) {
                cout << to_string(mat[i][j]) + " ";
            }
            cout << endl;
        }
        cout << endl;
    }
};

// Driver code
int main()
{
    int N = 9;
    int K = 20;
    Sudoku* sudoku = new Sudoku(N, K);
    sudoku->fillValues();
    sudoku->printSudoku();
    return 0;
}

// This code is contributed by Aarti_Rathi

Java

/* Java program for Sudoku generator  */
import java.lang.*;

public class Sudoku
{
    int[] mat[];
    int N; // number of columns/rows.
    int SRN; // square root of N
    int K; // No. Of missing digits

    // Constructor
    Sudoku(int N, int K)
    {
        this.N = N;
        this.K = K;

        // Compute square root of N
        Double SRNd = Math.sqrt(N);
        SRN = SRNd.intValue();

        mat = new int[N][N];
    }

    // Sudoku Generator
    public void fillValues()
    {
        // Fill the diagonal of SRN x SRN matrices
        fillDiagonal();

        // Fill remaining blocks
        fillRemaining(0, SRN);

        // Remove Randomly K digits to make game
        removeKDigits();
    }

    // Fill the diagonal SRN number of SRN x SRN matrices
    void fillDiagonal()
    {

        for (int i = 0; i<N; i=i+SRN)

            // for diagonal box, start coordinates->i==j
            fillBox(i, i);
    }

    // Returns false if given 3 x 3 block contains num.
    boolean unUsedInBox(int rowStart, int colStart, int num)
    {
        for (int i = 0; i<SRN; i++)
            for (int j = 0; j<SRN; j++)
                if (mat[rowStart+i][colStart+j]==num)
                    return false;

        return true;
    }

    // Fill a 3 x 3 matrix.
    void fillBox(int row,int col)
    {
        int num;
        for (int i=0; i<SRN; i++)
        {
            for (int j=0; j<SRN; j++)
            {
                do
                {
                    num = randomGenerator(N);
                }
                while (!unUsedInBox(row, col, num));

                mat[row+i][col+j] = num;
            }
        }
    }

    // Random generator
    int randomGenerator(int num)
    {
        return (int) Math.floor((Math.random()*num+1));
    }

    // Check if safe to put in cell
    boolean CheckIfSafe(int i,int j,int num)
    {
        return (unUsedInRow(i, num) &&
                unUsedInCol(j, num) &&
                unUsedInBox(i-i%SRN, j-j%SRN, num));
    }

    // check in the row for existence
    boolean unUsedInRow(int i,int num)
    {
        for (int j = 0; j<N; j++)
           if (mat[i][j] == num)
                return false;
        return true;
    }

    // check in the row for existence
    boolean unUsedInCol(int j,int num)
    {
        for (int i = 0; i<N; i++)
            if (mat[i][j] == num)
                return false;
        return true;
    }

    // A recursive function to fill remaining
    // matrix
    boolean fillRemaining(int i, int j)
    {
        //  System.out.println(i+" "+j);
        if (j>=N && i<N-1)
        {
            i = i + 1;
            j = 0;
        }
        if (i>=N && j>=N)
            return true;

        if (i < SRN)
        {
            if (j < SRN)
                j = SRN;
        }
        else if (i < N-SRN)
        {
            if (j==(int)(i/SRN)*SRN)
                j =  j + SRN;
        }
        else
        {
            if (j == N-SRN)
            {
                i = i + 1;
                j = 0;
                if (i>=N)
                    return true;
            }
        }

        for (int num = 1; num<=N; num++)
        {
            if (CheckIfSafe(i, j, num))
            {
                mat[i][j] = num;
                if (fillRemaining(i, j+1))
                    return true;

                mat[i][j] = 0;
            }
        }
        return false;
    }

    // Remove the K no. of digits to
    // complete game
    public void removeKDigits()
    {
        int count = K;
        while (count != 0)
        {
            int cellId = randomGenerator(N*N)-1;

            // System.out.println(cellId);
            // extract coordinates i  and j
            int i = (cellId/N);
            int j = cellId%9;
            if (j != 0)
                j = j - 1;

            // System.out.println(i+" "+j);
            if (mat[i][j] != 0)
            {
                count--;
                mat[i][j] = 0;
            }
        }
    }

    // Print sudoku
    public void printSudoku()
    {
        for (int i = 0; i<N; i++)
        {
            for (int j = 0; j<N; j++)
                System.out.print(mat[i][j] + " ");
            System.out.println();
        }
        System.out.println();
    }

    // Driver code
    public static void main(String[] args)
    {
        int N = 9, K = 20;
        Sudoku sudoku = new Sudoku(N, K);
        sudoku.fillValues();
        sudoku.printSudoku();
    }
}

Python3

import random
import math

class Sudoku:
    def __init__(self, N, K):
        self.N = N
        self.K = K

        # Compute square root of N
        SRNd = math.sqrt(N)
        self.SRN = int(SRNd)
        self.mat = [[0 for _ in range(N)] for _ in range(N)]

    def fillValues(self):
        # Fill the diagonal of SRN x SRN matrices
        self.fillDiagonal()

        # Fill remaining blocks
        self.fillRemaining(0, self.SRN)

        # Remove Randomly K digits to make game
        self.removeKDigits()

    def fillDiagonal(self):
        for i in range(0, self.N, self.SRN):
            self.fillBox(i, i)

    def unUsedInBox(self, rowStart, colStart, num):
        for i in range(self.SRN):
            for j in range(self.SRN):
                if self.mat[rowStart + i][colStart + j] == num:
                    return False
        return True

    def fillBox(self, row, col):
        num = 0
        for i in range(self.SRN):
            for j in range(self.SRN):
                while True:
                    num = self.randomGenerator(self.N)
                    if self.unUsedInBox(row, col, num):
                        break
                self.mat[row + i][col + j] = num

    def randomGenerator(self, num):
        return math.floor(random.random() * num + 1)

    def checkIfSafe(self, i, j, num):
        return (self.unUsedInRow(i, num) and self.unUsedInCol(j, num) and self.unUsedInBox(i - i % self.SRN, j - j % self.SRN, num))

    def unUsedInRow(self, i, num):
        for j in range(self.N):
            if self.mat[i][j] == num:
                return False
        return True

    def unUsedInCol(self, j, num):
        for i in range(self.N):
            if self.mat[i][j] == num:
                return False
        return True


    def fillRemaining(self, i, j):
        # Check if we have reached the end of the matrix
        if i == self.N - 1 and j == self.N:
            return True

        # Move to the next row if we have reached the end of the current row
        if j == self.N:
            i += 1
            j = 0

        # Skip cells that are already filled
        if self.mat[i][j] != 0:
            return self.fillRemaining(i, j + 1)

        # Try filling the current cell with a valid value
        for num in range(1, self.N + 1):
            if self.checkIfSafe(i, j, num):
                self.mat[i][j] = num
                if self.fillRemaining(i, j + 1):
                    return True
                self.mat[i][j] = 0

        # No valid value was found, so backtrack
        return False

    def removeKDigits(self):
        count = self.K

        while (count != 0):
            i = self.randomGenerator(self.N) - 1
            j = self.randomGenerator(self.N) - 1
            if (self.mat[i][j] != 0):
                count -= 1
                self.mat[i][j] = 0

        return

    def printSudoku(self):
        for i in range(self.N):
            for j in range(self.N):
                print(self.mat[i][j], end=" ")
            print()

# Driver code
if __name__ == "__main__":
    N = 9
    K = 40
    sudoku = Sudoku(N, K)
    sudoku.fillValues()
    sudoku.printSudoku()

C#

/* C# program for Sudoku generator  */
using System;

public class Sudoku
{
    int[,] mat;
    int N; // number of columns/rows.
    int SRN; // square root of N
    int K; // No. Of missing digits

    // Constructor
    public Sudoku(int N, int K)
    {
        this.N = N;
        this.K = K;

        // Compute square root of N
        double SRNd = Math.Sqrt(N);
        SRN = (int)SRNd;

        mat = new int[N,N];
    }

    // Sudoku Generator
    public void fillValues()
    {
        // Fill the diagonal of SRN x SRN matrices
        fillDiagonal();

        // Fill remaining blocks
        fillRemaining(0, SRN);

        // Remove Randomly K digits to make game
        removeKDigits();
    }

    // Fill the diagonal SRN number of SRN x SRN matrices
    void fillDiagonal()
    {

        for (int i = 0; i<N; i=i+SRN)

            // for diagonal box, start coordinates->i==j
            fillBox(i, i);
    }

    // Returns false if given 3 x 3 block contains num.
    bool unUsedInBox(int rowStart, int colStart, int num)
    {
        for (int i = 0; i<SRN; i++)
            for (int j = 0; j<SRN; j++)
                if (mat[rowStart+i,colStart+j]==num)
                    return false;

        return true;
    }

    // Fill a 3 x 3 matrix.
    void fillBox(int row,int col)
    {
        int num;
        for (int i=0; i<SRN; i++)
        {
            for (int j=0; j<SRN; j++)
            {
                do
                {
                    num = randomGenerator(N);
                }
                while (!unUsedInBox(row, col, num));

                mat[row+i,col+j] = num;
            }
        }
    }

    // Random generator
    int randomGenerator(int num)
    {
        Random rand = new Random();
        return (int) Math.Floor((double)(rand.NextDouble()*num+1));
    }

    // Check if safe to put in cell
    bool CheckIfSafe(int i,int j,int num)
    {
        return (unUsedInRow(i, num) &&
                unUsedInCol(j, num) &&
                unUsedInBox(i-i%SRN, j-j%SRN, num));
    }

    // check in the row for existence
    bool unUsedInRow(int i,int num)
    {
        for (int j = 0; j<N; j++)
           if (mat[i,j] == num)
                return false;
        return true;
    }

    // check in the row for existence
    bool unUsedInCol(int j,int num)
    {
        for (int i = 0; i<N; i++)
            if (mat[i,j] == num)
                return false;
        return true;
    }

    // A recursive function to fill remaining
    // matrix
    bool fillRemaining(int i, int j)
    {
        //  System.out.println(i+" "+j);
        if (j>=N && i<N-1)
        {
            i = i + 1;
            j = 0;
        }
        if (i>=N && j>=N)
            return true;

        if (i < SRN)
        {
            if (j < SRN)
                j = SRN;
        }
        else if (i < N-SRN)
        {
            if (j==(int)(i/SRN)*SRN)
                j =  j + SRN;
        }
        else
        {
            if (j == N-SRN)
            {
                i = i + 1;
                j = 0;
                if (i>=N)
                    return true;
            }
        }

        for (int num = 1; num<=N; num++)
        {
            if (CheckIfSafe(i, j, num))
            {
                mat[i,j] = num;
                if (fillRemaining(i, j+1))
                    return true;

                mat[i,j] = 0;
            }
        }
        return false;
    }

    // Remove the K no. of digits to
    // complete game
    public void removeKDigits()
    {
        int count = K;
        while (count != 0)
        {
            int cellId = randomGenerator(N*N)-1;

            // System.out.println(cellId);
            // extract coordinates i  and j
            int i = (cellId/N);
            int j = cellId%9;
            if (j != 0)
                j = j - 1;

            // System.out.println(i+" "+j);
            if (mat[i,j] != 0)
            {
                count--;
                mat[i,j] = 0;
            }
        }
    }

    // Print sudoku
    public void printSudoku()
    {
        for (int i = 0; i<N; i++)
        {
            for (int j = 0; j<N; j++)
                Console.Write(mat[i,j] + " ");
            Console.WriteLine();
        }
        Console.WriteLine();
    }

    // Driver code
    public static void Main(string[] args)
    {
        int N = 9, K = 20;
        Sudoku sudoku = new Sudoku(N, K);
        sudoku.fillValues();
        sudoku.printSudoku();
    }
}

// This code is contributed by rrrtnx.

Javascript

// JS program to implement the approach
class Sudoku {

    // Constructor
    constructor(N, K) {
        this.N = N;
        this.K = K;

        // Compute square root of N
        const SRNd = Math.sqrt(N);
        this.SRN = Math.floor(SRNd);

        // Initialize all entries as false to indicate
        // that there are no edges initially
        this.mat = Array.from({
            length: N
        }, () => Array.from({
            length: N
        }, () => 0));
    }

    // Sudoku Generator
    fillValues() {
        // Fill the diagonal of SRN x SRN matrices
        this.fillDiagonal();

        // Fill remaining blocks
        this.fillRemaining(0, this.SRN);

        // Remove Randomly K digits to make game
        this.removeKDigits();
    }

    // Fill the diagonal SRN number of SRN x SRN matrices
    fillDiagonal() {
        for (let i = 0; i < this.N; i += this.SRN) {
            // for diagonal box, start coordinates->i==j
            this.fillBox(i, i);
        }
    }

    // Returns false if given 3 x 3 block contains num.
    unUsedInBox(rowStart, colStart, num) {
        for (let i = 0; i < this.SRN; i++) {
            for (let j = 0; j < this.SRN; j++) {
                if (this.mat[rowStart + i][colStart + j] === num) {
                    return false;
                }
            }
        }
        return true;
    }

    // Fill a 3 x 3 matrix.
    fillBox(row, col) {
        let num = 0;
        for (let i = 0; i < this.SRN; i++) {
            for (let j = 0; j < this.SRN; j++) {
                while (true) {
                    num = this.randomGenerator(this.N);
                    if (this.unUsedInBox(row, col, num)) {
                        break;
                    }
                }
                this.mat[row + i][col + j] = num;
            }
        }
    }

    // Random generator
    randomGenerator(num) {
        return Math.floor(Math.random() * num + 1);
    }

    // Check if safe to put in cell
    checkIfSafe(i, j, num) {
        return (
            this.unUsedInRow(i, num) &&
            this.unUsedInCol(j, num) &&
            this.unUsedInBox(i - (i % this.SRN), j - (j % this.SRN), num)
        );
    }

    // check in the row for existence
    unUsedInRow(i, num) {
        for (let j = 0; j < this.N; j++) {
            if (this.mat[i][j] === num) {
                return false;
            }
        }
        return true;
    }

    // check in the row for existence
    unUsedInCol(j, num) {
        for (let i = 0; i < this.N; i++) {
            if (this.mat[i][j] === num) {
                return false;
            }
        }
        return true;
    }

    // A recursive function to fill remaining
    // matrix
    fillRemaining(i, j) {
        // Check if we have reached the end of the matrix
        if (i === this.N - 1 && j === this.N) {
            return true;
        }

        // Move to the next row if we have reached the end of the current row
        if (j === this.N) {
            i += 1;
            j = 0;
        }


        // Skip cells that are already filled
        if (this.mat[i][j] !== 0) {
            return this.fillRemaining(i, j + 1);
        }

        // Try filling the current cell with a valid value
        for (let num = 1; num <= this.N; num++) {
            if (this.checkIfSafe(i, j, num)) {
                this.mat[i][j] = num;
                if (this.fillRemaining(i, j + 1)) {
                    return true;
                }
                this.mat[i][j] = 0;
            }
        }

        // No valid value was found, so backtrack
        return false;
    }

    // Print sudoku
    printSudoku() {
        for (let i = 0; i < this.N; i++) {
                console.log(this.mat[i].join(" "))
        }
    }

    // Remove the K no. of digits to
    // complete game
    removeKDigits() {
        let count = this.K;

        while (count !== 0) {
            // extract coordinates i and j
            let i = Math.floor(Math.random() * this.N);
            let j = Math.floor(Math.random() * this.N);
            if (this.mat[i][j] !== 0) {
                count--;
                this.mat[i][j] = 0;
            }
        }

        return;
    }
}


// Driver code
let N = 9
let K = 40
let sudoku = new Sudoku(N, K)
sudoku.fillValues()
sudoku.printSudoku()

// This code is contributed by phasing17.

Output: [0 means not filled] 

2 3 0 4 1 5 0 6 8 
0 8 0 2 3 6 5 1 9 
1 6 0 9 8 7 2 3 4 
3 1 7 0 9 4 0 2 5 
4 5 8 1 2 0 6 9 7 
9 2 6 0 5 8 3 0 1 
0 0 0 5 0 0 1 0 2 
0 0 0 8 4 2 9 0 3 
5 9 2 3 7 1 4 8 6 

Time Complexity: O(N²) 
Auxiliary Space: O(N²), since N² extra space has been taken.

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