Video Note

Sudoku

fixed cells free cells

7.7 Problem: Sudoku

This book teaches how to program using a wide variety of problems with various levels of difficulty. We use simple, short, and stimulating examples to introduce programming and problem-solving techniques and use interesting and challenging examples to motivate students. This section presents an interesting problem of a sort that appears in the newspaper every day. It is a number-placement puzzle, commonly known as Sudoku. This is a very challenging problem. To make it accessible to the novice, this section presents a solution to a simplified version of the Sudoku problem, which is to verify whether a solution is correct. The complete solution for solving the Sudoku problem is presented in Supplement VII.A.

Sudoku is a 9 * 9 grid divided into smaller 3 * 3 boxes (also called regions or blocks), as shown in Figure 7.4(a). Some cells, called fixed cells, are populated with numbers from 1 to 9. The objective is to fill the empty cells, also called free cells, with numbers 1 to 9 so that every row, every column, and every 3 * 3 box contains the numbers 1 to 9, as shown in Figure 7.4(b).

A solution grid is

{{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} };

FIGURE 7.6 A solution is stored in grid.

A simplified version of the Sudoku problem is to check the validity of a solution. The program in Listing 7.4 prompts the user to enter a solution and reports whether it is valid.

LISTING 7.4 CheckSudokuSolution.java

1 import java.util.Scanner;

2

3 public class CheckSudokuSolution {

4 public static void main(String[] args) {

5// Read a Sudoku solution

6 int[][] grid = readASolution();

7

8 System.out.println(isValid(grid) ? "Valid solution" :

9 "Invalid solution");

10 }

11

12/** Read a Sudoku solution from the console */

13public static int[][] readASolution() {

14// Create a Scanner

15Scanner input = new Scanner(System.in);

read input

solution valid?

read solution

74if (temp[i] != i + 1)

75return false;

76

77return true; // The list contains exactly 1 to 9

78}

79}

Enter a Sudoku puzzle solution: 9 6 3 1 7 4 2 5 8 1 7 8 3 2 5 6 4 9 2 5 4 6 8 9 7 3 1 8 2 1 4 3 7 5 9 6 4 9 6 8 5 2 3 1 7 7 3 5 9 6 1 8 2 4 5 8 9 7 1 3 4 6 2 3 1 7 2 4 6 9 8 5 6 4 2 5 9 8 1 7 3

Valid solution

The program invokes the readASolution() method (line 6) to read a Sudoku solution and return a two-dimensional array representing a Sudoku grid.

The isValid(grid) method (lines 27–61) checks whether every row contains numbers 1 to 9 (lines 29–31). grid is a two-dimensional array. grid[i] is a one-dimensional array for the ith row. Invoking is1To9(grid[i]) returns true if the row grid[i] contains exactly numbers from 1 to 9 (line 30).

To check whether each column in grid has numbers 1 to 9, get a column into a onedimensional array (lines 36–39) and invoke the is1To9 method to check whether it has 1 to 9 (line 41).

To check whether each small 3 * 3 box in grid has numbers 1 to 9, get a box into a onedimensional array (lines 49–53) and invoke the is1To9 method to check whether it has 1 to 9 (line 55).

How do you locate all the cells in the same box? First, locate the starting cells of the 3 * 3 boxes. They are at (3i, 3j) for i = 0, 1, 2 and j = 0, 1, 2, as illustrated in Figure 7.7.

isValid method check rows

check columns

check small boxes