## Introduction

Sudoku is a logic-based number-placement puzzle. The objective is to fill a 9×9 grid so that each column, each row, and each of the nine 3×3 boxes (also called blocks or regions) contains the digits from 1 to 9 only one time each. The puzzle provides a partially completed grid (http://en.wikipedia.org/wiki/Sudoku). The following is an example of a Sudoku puzzle:

SUDOKU is a Gendex module for solving/generating Sudoku puzzle. It uses an unpublished exchange algorithm of its author and can solve Sudoku puzzles of different grid sizes: 4x4, 6x6, 8x8 and 9x9 (see http://www.menneske.no/sudoku/eng/). The box sizes which correspond to each mentioned grid size are 2x2, 2x3, 2x4 and 3x3 respectively.

For a given Sudoku puzzle of grid size *nxn*, the mentioned exchange
algorithm minimizes the objective function *f* where
*f*=*F*-*n ^{2}* and

*F*is the sum of the number of times each digit occur with itself and other digits having the same values in each row and in each column. Users who are familiar with design theory can see that

*F*is sum of the trace of

*NN'*(concurrence matrix of column components of the grid) and trace of

*MM'*(concurrence matrix of row components of the grid). The 2-exchange algorithm swaps the positions of two digits in a replicate. The 3-exchange algorithm swaps the positions of three digits in a replicate. When

*f*becomes 0, the solution is obtained.

**Note**: SUDOKU is designed to solve a Sudoku puzzle of any
size. However, it is much more successful with puzzles of size 9x9 or less.

## Using SUDOKU

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to find the solution for Sudoku puzzle in the previous Section. At the working directory, create a file say s9x9.txt with missing cells replaced by 0's.

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

At the working directory, type the following command at the command prompt (case is important):

java -cp c:\gendex SUDOKU

The SUDOKU window will pop up. Enter s9x9.txt in the **File** field.
Note that the default random seed is the
one obtained from the system clock and the default number of tries is 100000.
You can change these default values if you wish to. The default algorithm is
3-exchange (which includes both 2- and 3-exchange). Again, you can change this
default setting. Now, click START, SUDOKU will start running and after try 87,
the solution for the puzzle pops up (as *f* value reaches 0) and SUDOKU stops:

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

Its solution given by SODOKU is given below:

To generate a Sudoku puzzle, you have to leave the **File**
field blank and specify (i) the grid size (either 9x9, 8x8, 6x6 or 4x4) and the
number of empty cells. Below is a generated Sudoku puzzle for the gird size 9x9
with 30 missing cells.

## Output

When SUDOKU is used to solve a Sudoku puzzle, the SUDOKU output screen contains the result of the best try. Information for this try includes:

- Try number;
- The number of iterations;
*f*, the objective function. The program automatically stops when*f*=0.- The random seed used;
- The Sudoku solution;
- The time in seconds SUDOKU used to obtain this solution.

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