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## Solving the Sudoku Using Integer Programming

The 9 X9 SUDOKU puzzle has the following rules. Each row and column must contain numbers from 1 to 9, and each of the inner boxes must contain numbers from 1 to 9. Each number must occur only once in each column and row and in each small box.

Let Xijk be set to 1 for all values of i, j, and k from 1 to 9. If cell (I,j) contains the number k, where I, j, and k all range from 1 to 9. The ith row and j denote the jth column, and k denotes an integer between 1 and 9. When x134 = 1, it means that the cell (1,3) contains the number 4. Row 1 or column 3 can be equal to 4 except cell (1,3).

We will use a total of 729 variables to model SUDOKU.

Now let’s formulate each of the three classes of rules algebraically.

Each line must contain a number between 1 and 9 only once.

For the first row, this rule (called “Constraint” in Integer Programming language) looks like this.

for each I from 1 to 9 and for each k from 1 to 9 (I is the mathematical representation of the counter variable)

sum (Xijk) = 1 for all j from 1 to 9;

For each number from 1 to 9, it is written in detail for the 1st row

X111 + X121 + X131 + X141 + X151 + X161 + X171 + X181 + X191 = 1.

X112 + X122 + X132 + X142 + X152 + X162 + X172 + X182 + X192 = 1.

X113 + X123 + X133 + X143 + X153 + X163 + X173 + X183 + X193 = 1.

X114 + X124 + X134 + X144 + X154 + X164 + X174 + X184 + X194 = 1.

These equations are valid for variables X115 through X119.

Similarly, let’s form equations for the rules for every number between 1 and 9 that occurs only once in each of the 9 columns.

Written in mathematical notation,

Sum of X for each j from 1 to 9 (for all I and k from 1 to 9) = 1

Each number between 1 and 9 is written in detail for several columns

Column 1

X111 + X211 + X311 + X411 + X511 + X611 + X711 + X811 + X911 = 1.

X112 + X212 + X312 + X412 + X512 + X612 + X712 + X812 + X912 = 1.

X113 + X213 + X313 + X413 + X513 + X613 + X713 + X813 + X913 = 1.

This must be filled in for all other numbers from 4 to 9.

Column 2

X121 + X221 + X321 + X421 + X521 + X621 + X721 + X821 + X921 = 1.

X122 + X222 + X322 + X422 + X522 + X622 + X722 + X822 + X922 = 1.

X123 + X223 + X323 + X423 + X523 + X623 + X723 + X823 + X923 = 1.

This must be filled in for all other numbers from 4 to 9.

Now let’s represent the small boxes (3 x 3) with a total number of 9 squares.

So every 3 x 3 square has only one 1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9 and so on. should be

Cells Columns (1 to 3) and Rows (1 to 3) Columns (4 to 6) and Rows (1 to 3) Columns (7 to 9) to) lines (1 to 3). For the same set of columns, they occur in rows (4 to 6) and (6 to 9). So let’s do the equations for just one small square between the columns (1 through 3) and the rows (1 through 3). Corresponding decision variables for the digit “1” (9 in total)

X111, X121, X131, X211, X221, X231, X311, X321, X331.

Let’s form an equation where there is only one “1” in this (3 x3) square.

So the equation is:

X111 + X121+ X131 + X211 +X221+ X231+ X311 + X321 + X331 = 1.

The above equation implies that only one of these 9 variables or only one of these nine cells can take the value 1.

Similarly, the limits should be drawn up to 9 for the number “2”, and for the number “3”.

In addition to the equations describing the constraints for integer programming problems, there must also be integer constraints placed on each variable so that when the resulting system of equations is solved, the solution to the Xijk variable is either 0 or 1. .

The geometric equivalent of a linear programming problem with an objective function and some constraints is nothing but a dimensional polyhedron, where n represents the number of constraints in the problem. Usually the optimal solution is found at the vertices of the polytope, also the rules of some methods like SIMPLEX require the polytope to be convex so that it is possible to traverse from vertex to vertex along the edges to find the optimal solution.

Additionally, imposing integer constraints would mean that the optimal solution would not be at the vertices of the polytope, since the solution found at a vertex might not be an integer. Thus, given that the optimal solution must be either 0 or 1, geometrically the solution will lie inside the feasible region of the convex polytope and on one of the many straight lines arising from the hypertop equivalent to the integer Xi jk. values.

Note that the above solution used 729 decision variables and 81 row constraints. 81 column constraints and 729 least squares constraints for a total of 901 constraints. There can be many objective functions, but one objective function (the sum of all 729 variables) can be formulated as finding the min. The number of restrictions can be reduced by finding some resources.

These equations above cannot be solved using programming languages like Visual Basic, Pascal or C. Integer programming problems CPLEX optimizer, Excel add-in for solving Linear Programming problems, Lingo, etc. can be solved using optimization programs such as

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