Mastering Sudoku with Integer Programming in GAMS: A Beginner's Guide

Sudoku, the popular logic-based number-placement puzzle, has captivated puzzle enthusiasts worldwide. While many enjoy solving Sudoku puzzles by hand, the challenge can be elevated by employing mathematical optimization techniques. In this article, we'll explore how to create a Sudoku solver using Integer Programming (IP) in GAMS, a powerful mathematical modeling language. We'll delve into the strategy, the GAMS setup, and the gameplay to help you master this intriguing combination of logic and mathematics.

Understanding Sudoku

Sudoku is a grid-based, number-placement puzzle that consists of a 9x9 grid divided into nine 3x3 subgrids called "regions." The objective is to fill the grid with digits so that each column, each row, and each of the nine 3x3 subgrids contain all of the digits from 1 to 9. The key to solving Sudoku is logical deduction and elimination.

Introduction to Integer Programming in GAMS

Integer Programming (IP) is a mathematical optimization technique that solves problems where some or all of the variables are restricted to be integers. GAMS (General Algebraic Modeling System) is a high-level modeling language for mathematical optimization problems. It allows users to define and solve complex optimization problems efficiently.

Setting Up GAMS for Sudoku

To start, you'll need to have GAMS installed on your computer. Once you have GAMS, follow these steps to set up a Sudoku solver:

1. Define the Model: Start by defining the model using GAMS syntax. This includes setting up the sets, parameters, and decision variables.
2. Constraints: Implement constraints that ensure each row, column, and region contains the digits 1 through 9 without repetition.
3. Objective Function: The objective function is typically set to minimize or maximize a value, though in Sudoku, it's more about finding a valid solution rather than optimizing a value.

Writing the GAMS Code

Here's a simplified example of the GAMS code for a Sudoku solver:

Sets
    i /1*9/    "rows"
    j /1*9/    "columns"
    k /1*9/    "digits"
    region(i,j) /1*3,1*3/ "3x3 regions"
    cells(i,j) /1*9,1*9/ "all cells in the grid";

Parameters
    region(i,j) /1.1,1.1,1.2,1.2,1.3,1.3,2.1,2.1,2.2,2.2,2.3,2.3,
                  3.1,3.1,3.2,3.2,3.3,3.3/;

Positive Variables
    x(i,j,k) "solution variable";

Equations
    constraint_row(i,j,k) "constraints for rows"
    constraint_column(i,j,k) "constraints for columns"
    constraint_region(i,j,k) "constraints for regions";

constraint_row(i,j,k).. x(i,j,k) =l= 1;
constraint_column(i,j,k).. x(i,j,k) =l= 1;
constraint_region(i,j,k).. x(i,j,k) =l= 1;

Model sudoku /all/;

Solve sudoku using mip minimizing 0;

Display x.l;

Playing the Sudoku Game

After running the GAMS code, you will get the solution for the Sudoku puzzle. The x.l values represent the digits to be placed in each cell. You can then use this solution to play the puzzle or use it as a reference for manual solving.

Conclusion

By combining the logical challenge of Sudoku with the power of Integer Programming in GAMS, you can gain a deeper understanding of both. This guide provides a starting point for creating your own Sudoku solver, allowing you to explore the intricate world of mathematical optimization. Happy solving!