Solving LinkedIn Mini Sudoku with Linear Optimization

What Sudoku is¶

Mini Sudoku is based on the classic Sudoku game (from the Japanese 数独, meaning unique numbers in a free translation), which originally consists of a 9 $\times$ 9 grid and 9 smaller 3 $\times$ 3 matrices, with some squares already pre-filled with digits from 1 to 9. The meaning of its name is due to its objective of filling all the squares without repeating digits in the rows, columns, and smaller grids.

How to Play Mini Sudoku¶

In the case of Mini Sudoku, the game consists of a grid with smaller dimensions than the classic, 6 $\times$ 6 grid, composed of six 2 $\times$ 3 submatrices.

Objective

Fill all the empty spaces in the game grid with digits from 1 to 6

Rules

Each row, column, and mini-grid of the main game matrix must be filled with a digit from 1 to 6, without repetition in each row, column, or smaller grid.

Problem Modeling¶

As I did in my previous articles, the LO model for the Mini Sudoku game requires the definition of the following components:

• Ranges

• Sets

• Objective function

• Decision Variables

• Constraints

First, let’s define these components considering the most general scenario for building the Abstract Model for the Sudoku game before defining them for the more specific case of Mini Sudoku.

Ranges¶

In order to consider the most general cases, five ranges will be considered: $I$ and $J$ to represent the dimensions of the main grid, $U$ and $V$ for the dimensions of the smaller grids, and an interval $K$ for the range of possible values a square can receive.

$I = \{1, \cdots, n\}$

The row range, where $n$ is the total number of rows (in this case, $n = 6$)

$J = \{1, \cdots, n\}$

The column range, where the total number of columns is equal to the number of rows for dealing with a square matrix

$K = \{1, \cdots, n\}$

Range of possible values, where $n$ is the total number of possible digits, which is expected to be equal to the $n$ dimensions of the grid

$U = \{1, \cdots, p\}$

Number of rows in the game’s submatrices, where $p$ is the total number of rows in each submatrix

$V = \{1, \cdots, q\}$

Number of columns in the game’s submatrices, where $q$ is the total number of columns in each submatrix, which is expected to be $pq = n$, that is, the number of squares in each submatrix should be equal to the number of possible digits.

Sets¶

To facilitate the definition of the constraints, it is important to define at least the set $S$ of submatrices and $F$ of pre-filled squares. Furthermore, it is necessary to clearly define which squares of the board comprise each of the submatrices $S_{vu}$. For example, the first submatrix $S_{11}$ is composed of squares in rows 1 and 2 and columns whose index ranges go from 1 to 3. $S_{12}$ is also composed of squares in rows 1 and 2, but the column indices range from 4 to 6, which is the second half of the column range; and so on for the remaining $S_{vu}$.

$S = \{S_{vu} \mid \forall v \in V, \forall u \in U\}$

Set of submatrices $S_{vu}$ existing in the game

$S_{vu} = \{(i, j) \mid \forall i \in \{p(v-1)+1, \cdots, pv\}, \forall j \in \{q(u-1)+1, \cdots, qu\}\}$

Set of squares $(i, j) \subseteq I \times J$ that belong to the submatrix $S_{vu}$

$F = \{(i, j, k) \mid i \in I, j \in J, k \in K\} \subseteq I \times J \times K$

Subset of pre-filled squares.

Decision Variables¶

The decision variables will be binary $x_{ijk}$, representing the decision of whether square $(i, j)$ is filled with the value $k$. Therefore, as with the models of the other LinkedIn minigames, the Mini Sudoku is a BLOP.

$x_{ijk} \in \{0,1\}, \forall (i,j,k) \in I \times J \times K$

$x_{ijk} = 1$ if square $(i,j)$ is filled with the digit $k$

$x_{ijk} = 0$ otherwise.

Objective Function¶

Once again, as in previous articles, the Sudoku optimization problem does not have a function to be optimized, since we only want to find a solution that satisfies all the rules of the game. Therefore, the BLPP is a feasibility problem, whose objective function consists of maximizing (or minimizing) an arbitrary constant.

Constraints¶

Finally, with all the previously defined components, let’s translate the Sudoku rules into mathematical formulations for the BLOP model.

Binary Constraints

First of all, it’s important to clearly state the binary nature of the decision variables in the constraint set.

$$x_{ijk} \in \{0, 1\}, \forall (i, j, k) \in I \times J \times K$$

(2)

Unique-Digits-Per-Row Constraints

Since there can be no repetition of values for each row, the sum of the $x_{ijk}$ for each row $i$ must be equal to 1, and there must be a such constraint for column $j$ and for each digit $k$, which in the case of Mini Sudoku will result in 36 constraints.

$$\sum_{j \in J}{x_{ijk}}=1, \forall i \in I, \forall k \in K$$

(3)

Unique-Digits-Per-Column Constraints

The same logic applies to each column $j$ of the game, with one constraint for each row $i$ and possible digit $k$, resulting in 36 more constraints.

$$\sum_{i \in I}{x_{ijk}}=1, \forall j \in J, \forall k \in K$$

(4)

Unique-Digits-Per-Submatrix Constraints

With the set $S$ of submatrices already well defined, it becomes easier to define the set of constraints that prevent repetition of digits for each submatrix $S_{vu}$.

$$\sum_{(i,j) \in S_{vu}}{x_{ijk}}=1, \forall S_{vu} \in S, \forall k \in K$$

(5)

Single-Digit-Per-Square Constraints

In addition, it is necessary to impose a set of constraints to prevent a square from being filled with more than one digit, which is achieved if the sum of $x_{ijk}$ is equal to 1 for each square $(i, j)$ existing in the game; therefore, 36 more constraints in the case of Mini Sudoku.

$$\sum_{k \in K}{x_{ijk}}=1, \forall i \in I, \forall j \in J$$

(6)

Already-Filled-Squares Constraints

Finally, for each already filled square, we must remember to impose that $x_{ijk} = 1$ if the square $(i, j)$ is already filled with the digit $k$.

$$x_{ijk}=1, \forall (i,j,k) \in F$$

(7)

Abstract Model¶

With all the components set, we now have assembled the abstract model for a Sudoku game. It’s important to remember that this model assumes the game’s submatrices will be rectangular with dimensions $p \times q$, such that $pq = n$.

Concrete Model¶

The example to be solved in this notebook will be Mini Sudoku No. 60, published on LinkedIn on October 10^th, 2025

Based on the abstract model, it is possible to instantiate a concrete model for this game, as shown below.

S.t.:

Unique-Digits-Per-Row Constraints

$x_{111} + x_{121} + x_{131} + x_{141} + x_{151} + x_{161} = 1$ _{(Digit 1 on Row 1)}

$x_{211} + x_{221} + x_{231} + x_{241} + x_{251} + x_{261} = 1$ _{(Digit 1 on Row 2)}

$x_{311} + x_{321} + x_{331} + x_{341} + x_{351} + x_{361} = 1$ _{(Digit 1 on Row 3)}

$x_{411} + x_{421} + x_{431} + x_{441} + x_{451} + x_{461} = 1$ _{(Digit 1 on Row 4)}

$x_{511} + x_{521} + x_{531} + x_{541} + x_{551} + x_{561} = 1$ _{(Digit 1 on Row 5)}

$x_{611} + x_{621} + x_{631} + x_{641} + x_{651} + x_{661} = 1$ _{(Digit 1 on Row 6)}

$x_{112} + x_{122} + x_{132} + x_{142} + x_{152} + x_{162} = 1$ _{(Digit 2 on Row 1)}

$x_{212} + x_{222} + x_{232} + x_{242} + x_{252} + x_{262} = 1$ _{(Digit 2 on Row 2)}

$x_{312} + x_{322} + x_{332} + x_{342} + x_{352} + x_{362} = 1$ _{(Digit 2 on Row 3)}

$x_{412} + x_{422} + x_{432} + x_{442} + x_{452} + x_{462} = 1$ _{(Digit 2 on Row 4)}

$x_{512} + x_{522} + x_{532} + x_{542} + x_{552} + x_{562} = 1$ _{(Digit 2 on Row 5)}

$x_{612} + x_{622} + x_{632} + x_{642} + x_{652} + x_{662} = 1$ _{(Digit 2 on Row 6)}

$x_{113} + x_{123} + x_{133} + x_{143} + x_{153} + x_{163} = 1$ _{(Digit 3 on Row 1)}

$x_{213} + x_{223} + x_{233} + x_{243} + x_{253} + x_{263} = 1$ _{(Digit 3 on Row 2)}

$x_{313} + x_{323} + x_{333} + x_{343} + x_{353} + x_{363} = 1$ _{(Digit 3 on Row 3)}

$x_{413} + x_{423} + x_{433} + x_{443} + x_{453} + x_{463} = 1$ _{(Digit 3 on Row 4)}

$x_{513} + x_{523} + x_{533} + x_{543} + x_{553} + x_{563} = 1$ _{(Digit 3 on Row 5)}

$x_{613} + x_{623} + x_{633} + x_{643} + x_{653} + x_{663} = 1$ _{(Digit 3 on Row 6)}

$x_{114} + x_{124} + x_{134} + x_{144} + x_{154} + x_{164} = 1$ _{(Digit 4 on Row 1)}

$x_{214} + x_{224} + x_{234} + x_{244} + x_{254} + x_{264} = 1$ _{(Digit 4 on Row 2)}

$x_{314} + x_{324} + x_{334} + x_{344} + x_{354} + x_{364} = 1$ _{(Digit 4 on Row 3)}

$x_{414} + x_{424} + x_{434} + x_{444} + x_{454} + x_{464} = 1$ _{(Digit 4 on Row 4)}

$x_{514} + x_{524} + x_{534} + x_{544} + x_{554} + x_{564} = 1$ _{(Digit 4 on Row 5)}

$x_{614} + x_{624} + x_{634} + x_{644} + x_{654} + x_{664} = 1$ _{(Digit 4 on Row 6)}

$x_{115} + x_{125} + x_{135} + x_{145} + x_{155} + x_{165} = 1$ _{(Digit 5 on Row 1)}

$x_{215} + x_{225} + x_{235} + x_{245} + x_{255} + x_{265} = 1$ _{(Digit 5 on Row 2)}

$x_{315} + x_{325} + x_{335} + x_{345} + x_{355} + x_{365} = 1$ _{(Digit 5 on Row 3)}

$x_{415} + x_{425} + x_{435} + x_{445} + x_{455} + x_{465} = 1$ _{(Digit 5 on Row 4)}

$x_{515} + x_{525} + x_{535} + x_{545} + x_{555} + x_{565} = 1$ _{(Digit 5 on Row 5)}

$x_{615} + x_{625} + x_{635} + x_{645} + x_{655} + x_{665} = 1$ _{(Digit 5 on Row 6)}

$x_{116} + x_{126} + x_{136} + x_{146} + x_{156} + x_{166} = 1$ _{(Digit 6 on Row 1)}

$x_{216} + x_{226} + x_{236} + x_{246} + x_{256} + x_{266} = 1$ _{(Digit 6 on Row 2)}

$x_{316} + x_{326} + x_{336} + x_{346} + x_{356} + x_{366} = 1$ _{(Digit 6 on Row 3)}

$x_{416} + x_{426} + x_{436} + x_{446} + x_{456} + x_{466} = 1$ _{(Digit 6 on Row 4)}

$x_{516} + x_{526} + x_{536} + x_{546} + x_{556} + x_{566} = 1$ _{(Digit 6 on Row 5)}

$x_{616} + x_{626} + x_{636} + x_{646} + x_{656} + x_{666} = 1$ _{(Digit 6 on Row 6)}

Unique-Digits-Per-Column Constraints

$x_{111} + x_{211} + x_{311} + x_{411} + x_{511} + x_{611} = 1$ _{(Digit 1 on Column 1)}

$x_{121} + x_{221} + x_{321} + x_{421} + x_{521} + x_{621} = 1$ _{(Digit 1 on Column 2)}

$x_{131} + x_{231} + x_{331} + x_{431} + x_{531} + x_{631} = 1$ _{(Digit 1 on Column 3)}

$x_{141} + x_{241} + x_{341} + x_{441} + x_{541} + x_{641} = 1$ _{(Digit 1 on Column 4)}

$x_{151} + x_{251} + x_{351} + x_{451} + x_{551} + x_{651} = 1$ _{(Digit 1 on Column 5)}

$x_{161} + x_{261} + x_{361} + x_{461} + x_{561} + x_{661} = 1$ _{(Digit 1 on Column 6)}

$x_{112} + x_{212} + x_{312} + x_{412} + x_{512} + x_{612} = 1$ _{(Digit 2 on Column 1)}

$x_{122} + x_{222} + x_{322} + x_{422} + x_{522} + x_{622} = 1$ _{(Digit 2 on Column 2)}

$x_{132} + x_{232} + x_{332} + x_{432} + x_{532} + x_{632} = 1$ _{(Digit 2 on Column 3)}

$x_{142} + x_{242} + x_{342} + x_{442} + x_{542} + x_{642} = 1$ _{(Digit 2 on Column 4)}

$x_{152} + x_{252} + x_{352} + x_{452} + x_{552} + x_{652} = 1$ _{(Digit 2 on Column 5)}

$x_{162} + x_{262} + x_{362} + x_{462} + x_{562} + x_{662} = 1$ _{(Digit 2 on Column 6)}

$x_{113} + x_{213} + x_{313} + x_{413} + x_{513} + x_{613} = 1$ _{(Digit 3 on Column 1)}

$x_{123} + x_{223} + x_{323} + x_{423} + x_{523} + x_{623} = 1$ _{(Digit 3 on Column 2)}

$x_{133} + x_{233} + x_{333} + x_{433} + x_{533} + x_{633} = 1$ _{(Digit 3 on Column 3)}

$x_{143} + x_{243} + x_{343} + x_{443} + x_{543} + x_{643} = 1$ _{(Digit 3 on Column 4)}

$x_{153} + x_{253} + x_{353} + x_{453} + x_{553} + x_{653} = 1$ _{(Digit 3 on Column 5)}

$x_{163} + x_{263} + x_{363} + x_{463} + x_{563} + x_{663} = 1$ _{(Digit 3 on Column 6)}

$x_{114} + x_{214} + x_{314} + x_{414} + x_{514} + x_{614} = 1$ _{(Digit 4 on Column 1)}

$x_{124} + x_{224} + x_{324} + x_{424} + x_{524} + x_{624} = 1$ _{(Digit 4 on Column 2)}

$x_{134} + x_{234} + x_{334} + x_{434} + x_{534} + x_{634} = 1$ _{(Digit 4 on Column 3)}

$x_{144} + x_{244} + x_{344} + x_{444} + x_{544} + x_{644} = 1$ _{(Digit 4 on Column 4)}

$x_{154} + x_{254} + x_{354} + x_{454} + x_{554} + x_{654} = 1$ _{(Digit 4 on Column 5)}

$x_{164} + x_{264} + x_{364} + x_{464} + x_{564} + x_{664} = 1$ _{(Digit 4 on Column 6)}

$x_{115} + x_{215} + x_{315} + x_{415} + x_{515} + x_{615} = 1$ _{(Digit 5 on Column 1)}

$x_{125} + x_{225} + x_{325} + x_{425} + x_{525} + x_{625} = 1$ _{(Digit 5 on Column 2)}

$x_{135} + x_{235} + x_{335} + x_{435} + x_{535} + x_{635} = 1$ _{(Digit 5 on Column 3)}

$x_{145} + x_{245} + x_{345} + x_{445} + x_{545} + x_{645} = 1$ _{(Digit 5 on Column 4)}

$x_{155} + x_{255} + x_{355} + x_{455} + x_{555} + x_{655} = 1$ _{(Digit 5 on Column 5)}

$x_{165} + x_{265} + x_{365} + x_{465} + x_{565} + x_{665} = 1$ _{(Digit 5 on Column 6)}

$x_{116} + x_{216} + x_{316} + x_{416} + x_{516} + x_{616} = 1$ _{(Digit 6 on Column 1)}

$x_{126} + x_{226} + x_{326} + x_{426} + x_{526} + x_{626} = 1$ _{(Digit 6 on Column 2)}

$x_{136} + x_{236} + x_{336} + x_{436} + x_{536} + x_{636} = 1$ _{(Digit 6 on Column 3)}

$x_{146} + x_{246} + x_{346} + x_{446} + x_{546} + x_{646} = 1$ _{(Digit 6 on Column 4)}

$x_{156} + x_{256} + x_{356} + x_{456} + x_{556} + x_{656} = 1$ _{(Digit 6 on Column 5)}

$x_{166} + x_{266} + x_{366} + x_{466} + x_{566} + x_{666} = 1$ _{(Digit 6 on Column 6)}

Unique-Digits-Per-Submatrix Constraints

$x_{111} + x_{121} + x_{131} + x_{211} + x_{221} + x_{231} = 1$ _{(Digit 1 on Submatrix $S_{11}$)}

$x_{112} + x_{122} + x_{132} + x_{212} + x_{222} + x_{232} = 1$ _{(Digit 2 on Submatrix $S_{11}$)}

$x_{113} + x_{123} + x_{133} + x_{213} + x_{223} + x_{233} = 1$ _{(Digit 3 on Submatrix $S_{11}$)}

$x_{114} + x_{124} + x_{134} + x_{214} + x_{224} + x_{234} = 1$ _{(Digit 4 on Submatrix $S_{11}$)}

$x_{115} + x_{125} + x_{135} + x_{215} + x_{225} + x_{235} = 1$ _{(Digit 5 on Submatrix $S_{11}$)}

$x_{116} + x_{126} + x_{136} + x_{216} + x_{226} + x_{236} = 1$ _{(Digit 6 on Submatrix $S_{11}$)}

$x_{141} + x_{151} + x_{161} + x_{241} + x_{251} + x_{261} = 1$ _{(Digit 1 on Submatrix $S_{12}$)}

$x_{142} + x_{152} + x_{162} + x_{242} + x_{252} + x_{262} = 1$ _{(Digit 2 on Submatrix $S_{12}$)}

$x_{143} + x_{153} + x_{163} + x_{243} + x_{253} + x_{263} = 1$ _{(Digit 3 on Submatrix $S_{12}$)}

$x_{144} + x_{154} + x_{164} + x_{244} + x_{254} + x_{264} = 1$ _{(Digit 4 on Submatrix $S_{12}$)}

$x_{145} + x_{155} + x_{165} + x_{245} + x_{255} + x_{265} = 1$ _{(Digit 5 on Submatrix $S_{12}$)}

$x_{146} + x_{156} + x_{166} + x_{246} + x_{256} + x_{266} = 1$ _{(Digit 6 on Submatrix $S_{12}$)}

$x_{311} + x_{321} + x_{331} + x_{411} + x_{421} + x_{431} = 1$ _{(Digit 1 on Submatrix $S_{21}$)}

$x_{312} + x_{322} + x_{332} + x_{412} + x_{422} + x_{432} = 1$ _{(Digit 2 on Submatrix $S_{21}$)}

$x_{313} + x_{323} + x_{333} + x_{413} + x_{423} + x_{433} = 1$ _{(Digit 3 on Submatrix $S_{21}$)}

$x_{314} + x_{324} + x_{334} + x_{414} + x_{424} + x_{434} = 1$ _{(Digit 4 on Submatrix $S_{21}$)}

$x_{315} + x_{325} + x_{335} + x_{415} + x_{425} + x_{435} = 1$ _{(Digit 5 on Submatrix $S_{21}$)}

$x_{316} + x_{326} + x_{336} + x_{416} + x_{426} + x_{436} = 1$ _{(Digit 6 on Submatrix $S_{21}$)}

$x_{341} + x_{351} + x_{331} + x_{441} + x_{451} + x_{431} = 1$ _{(Digit 1 on Submatrix $S_{22}$)}

$x_{342} + x_{352} + x_{362} + x_{442} + x_{452} + x_{462} = 1$ _{(Digit 2 on Submatrix $S_{22}$)}

$x_{343} + x_{353} + x_{363} + x_{443} + x_{453} + x_{463} = 1$ _{(Digit 3 on Submatrix $S_{22}$)}

$x_{344} + x_{354} + x_{364} + x_{444} + x_{454} + x_{464} = 1$ _{(Digit 4 on Submatrix $S_{22}$)}

$x_{345} + x_{355} + x_{365} + x_{445} + x_{455} + x_{465} = 1$ _{(Digit 5 on Submatrix $S_{22}$)}

$x_{346} + x_{356} + x_{366} + x_{446} + x_{456} + x_{466} = 1$ _{(Digit 6 on Submatrix $S_{22}$)}

$x_{511} + x_{521} + x_{531} + x_{611} + x_{621} + x_{631} = 1$ _{(Digit 1 on Submatrix $S_{31}$)}

$x_{512} + x_{522} + x_{532} + x_{612} + x_{622} + x_{632} = 1$ _{(Digit 2 on Submatrix $S_{31}$)}

$x_{513} + x_{523} + x_{533} + x_{613} + x_{623} + x_{633} = 1$ _{(Digit 3 on Submatrix $S_{31}$)}

$x_{514} + x_{524} + x_{534} + x_{614} + x_{624} + x_{634} = 1$ _{(Digit 4 on Submatrix $S_{31}$)}

$x_{515} + x_{525} + x_{535} + x_{615} + x_{625} + x_{635} = 1$ _{(Digit 5 on Submatrix $S_{31}$)}

$x_{516} + x_{526} + x_{536} + x_{616} + x_{626} + x_{636} = 1$ _{(Digit 6 on Submatrix $S_{31}$)}

$x_{541} + x_{551} + x_{561} + x_{641} + x_{651} + x_{631} = 1$ _{(Digit 1 on Submatrix $S_{32}$)}

$x_{542} + x_{552} + x_{562} + x_{642} + x_{652} + x_{662} = 1$ _{(Digit 2 on Submatrix $S_{32}$)}

$x_{543} + x_{553} + x_{563} + x_{643} + x_{653} + x_{663} = 1$ _{(Digit 3 on Submatrix $S_{32}$)}

$x_{544} + x_{554} + x_{564} + x_{644} + x_{654} + x_{664} = 1$ _{(Digit 4 on Submatrix $S_{32}$)}

$x_{545} + x_{555} + x_{565} + x_{645} + x_{655} + x_{665} = 1$ _{(Digit 5 on Submatrix $S_{32}$)}

$x_{546} + x_{556} + x_{566} + x_{646} + x_{656} + x_{666} = 1$ _{(Digit 6 on Submatrix $S_{32}$)}

Single-Digit-Per-Square Constraints

$x_{111} + x_{112} + x_{113} + x_{114} + x_{115} + x_{116} = 1$ _{(Square (1, 1))}

$x_{121} + x_{122} + x_{123} + x_{124} + x_{125} + x_{126} = 1$ _{(Square (1, 2))}

$x_{131} + x_{132} + x_{133} + x_{134} + x_{135} + x_{136} = 1$ _{(Square (1, 3))}

$x_{141} + x_{142} + x_{143} + x_{144} + x_{145} + x_{146} = 1$ _{(Square (1, 4))}

$x_{151} + x_{152} + x_{153} + x_{154} + x_{155} + x_{156} = 1$ _{(Square (1, 5))}

$x_{161} + x_{162} + x_{163} + x_{164} + x_{165} + x_{166} = 1$ _{(Square (1, 6))}

$x_{211} + x_{212} + x_{213} + x_{214} + x_{215} + x_{216} = 1$ _{(Square (2, 1))}

$x_{221} + x_{222} + x_{223} + x_{224} + x_{225} + x_{226} = 1$ _{(Square (2, 2))}

$x_{231} + x_{232} + x_{233} + x_{234} + x_{235} + x_{236} = 1$ _{(Square (2, 3))}

$x_{241} + x_{242} + x_{243} + x_{244} + x_{245} + x_{246} = 1$ _{(Square (2, 4))}

$x_{251} + x_{252} + x_{253} + x_{254} + x_{255} + x_{256} = 1$ _{(Square (2, 5))}

$x_{261} + x_{262} + x_{263} + x_{264} + x_{265} + x_{266} = 1$ _{(Square (2, 6))}

$x_{311} + x_{312} + x_{313} + x_{314} + x_{315} + x_{316} = 1$ _{(Square (3, 1))}

$x_{321} + x_{322} + x_{323} + x_{324} + x_{325} + x_{326} = 1$ _{(Square (3, 2))}

$x_{331} + x_{332} + x_{333} + x_{334} + x_{335} + x_{336} = 1$ _{(Square (3, 3))}

$x_{341} + x_{342} + x_{343} + x_{344} + x_{345} + x_{346} = 1$ _{(Square (3, 4))}

$x_{351} + x_{352} + x_{353} + x_{354} + x_{355} + x_{356} = 1$ _{(Square (3, 5))}

$x_{361} + x_{362} + x_{363} + x_{364} + x_{365} + x_{366} = 1$ _{(Square (3, 6))}

$x_{411} + x_{412} + x_{413} + x_{414} + x_{415} + x_{416} = 1$ _{(Square (4, 1))}

$x_{421} + x_{422} + x_{423} + x_{424} + x_{425} + x_{426} = 1$ _{(Square (4, 2))}

$x_{431} + x_{432} + x_{433} + x_{434} + x_{435} + x_{436} = 1$ _{(Square (4, 3))}

$x_{441} + x_{442} + x_{443} + x_{444} + x_{445} + x_{446} = 1$ _{(Square (4, 4))}

$x_{451} + x_{452} + x_{453} + x_{454} + x_{455} + x_{456} = 1$ _{(Square (4, 5))}

$x_{461} + x_{462} + x_{463} + x_{464} + x_{465} + x_{466} = 1$ _{(Square (4, 6))}

$x_{511} + x_{512} + x_{513} + x_{514} + x_{515} + x_{516} = 1$ _{(Square (5, 1))}

$x_{521} + x_{522} + x_{523} + x_{524} + x_{525} + x_{526} = 1$ _{(Square (5, 2))}

$x_{531} + x_{532} + x_{533} + x_{534} + x_{535} + x_{536} = 1$ _{(Square (5, 3))}

$x_{541} + x_{542} + x_{543} + x_{544} + x_{545} + x_{546} = 1$ _{(Square (5, 4))}

$x_{551} + x_{552} + x_{553} + x_{554} + x_{555} + x_{556} = 1$ _{(Square (5, 5))}

$x_{561} + x_{562} + x_{563} + x_{564} + x_{565} + x_{566} = 1$ _{(Square (5, 6))}

$x_{611} + x_{612} + x_{613} + x_{614} + x_{615} + x_{616} = 1$ _{(Square (6, 1))}

$x_{621} + x_{622} + x_{623} + x_{624} + x_{625} + x_{626} = 1$ _{(Square (6, 2))}

$x_{631} + x_{632} + x_{633} + x_{634} + x_{635} + x_{636} = 1$ _{(Square (6, 3))}

$x_{641} + x_{642} + x_{643} + x_{644} + x_{645} + x_{646} = 1$ _{(Square (6, 4))}

$x_{651} + x_{652} + x_{653} + x_{654} + x_{655} + x_{656} = 1$ _{(Square (6, 5))}

$x_{661} + x_{662} + x_{663} + x_{664} + x_{665} + x_{666} = 1$ _{(Square (6, 6))}

Already-Filled-Squares Constraints

$x_{111} = 1$ _{(Square (1, 1) has Digit 1)}

$x_{222} = 1$ _{(Square (2, 2) has Digit 2)}

$x_{253} = 1$ _{(Square (2, 5) has Digit 3)}

$x_{346} = 1$ _{(Square (3, 4) has Digit 6)}

$x_{435} = 1$ _{(Square (4, 3) has Digit 5)}

$x_{444} = 1$ _{(Square (4, 4) has Digit 4)}

$x_{524} = 1$ _{(Square (5, 2) has Digit 4)}

$x_{555} = 1$ _{(Square (5, 5) has Digit 5)}

$x_{666} = 1$ _{(Square (6, 6) has Digit 6)}

Binary Constraints

$x_{111} \in \{0,1\}$

$x_{112} \in \{0,1\}$

$x_{113} \in \{0,1\}$

$x_{114} \in \{0,1\}$

$x_{115} \in \{0,1\}$

$x_{116} \in \{0,1\}$

$x_{121} \in \{0,1\}$

$x_{122} \in \{0,1\}$

$x_{123} \in \{0,1\}$

$x_{124} \in \{0,1\}$

$x_{125} \in \{0,1\}$

$x_{126} \in \{0,1\}$

$x_{131} \in \{0,1\}$

$x_{132} \in \{0,1\}$

$x_{133} \in \{0,1\}$

$x_{134} \in \{0,1\}$

$x_{135} \in \{0,1\}$

$x_{136} \in \{0,1\}$

$x_{141} \in \{0,1\}$

$x_{142} \in \{0,1\}$

$x_{143} \in \{0,1\}$

$x_{144} \in \{0,1\}$

$x_{145} \in \{0,1\}$

$x_{146} \in \{0,1\}$

$x_{151} \in \{0,1\}$

$x_{152} \in \{0,1\}$

$x_{153} \in \{0,1\}$

$x_{154} \in \{0,1\}$

$x_{155} \in \{0,1\}$

$x_{156} \in \{0,1\}$

$x_{161} \in \{0,1\}$

$x_{162} \in \{0,1\}$

$x_{163} \in \{0,1\}$

$x_{164} \in \{0,1\}$

$x_{165} \in \{0,1\}$

$x_{166} \in \{0,1\}$

$x_{211} \in \{0,1\}$

$x_{212} \in \{0,1\}$

$x_{213} \in \{0,1\}$

$x_{214} \in \{0,1\}$

$x_{215} \in \{0,1\}$

$x_{216} \in \{0,1\}$

$x_{221} \in \{0,1\}$

$x_{222} \in \{0,1\}$

$x_{223} \in \{0,1\}$

$x_{224} \in \{0,1\}$

$x_{225} \in \{0,1\}$

$x_{226} \in \{0,1\}$

$x_{231} \in \{0,1\}$

$x_{232} \in \{0,1\}$

$x_{233} \in \{0,1\}$

$x_{234} \in \{0,1\}$

$x_{235} \in \{0,1\}$

$x_{236} \in \{0,1\}$

$x_{241} \in \{0,1\}$

$x_{242} \in \{0,1\}$

$x_{243} \in \{0,1\}$

$x_{244} \in \{0,1\}$

$x_{245} \in \{0,1\}$

$x_{246} \in \{0,1\}$

$x_{251} \in \{0,1\}$

$x_{252} \in \{0,1\}$

$x_{253} \in \{0,1\}$

$x_{254} \in \{0,1\}$

$x_{255} \in \{0,1\}$

$x_{256} \in \{0,1\}$

$x_{261} \in \{0,1\}$

$x_{262} \in \{0,1\}$

$x_{263} \in \{0,1\}$

$x_{264} \in \{0,1\}$

$x_{265} \in \{0,1\}$

$x_{266} \in \{0,1\}$

$x_{311} \in \{0,1\}$

$x_{312} \in \{0,1\}$

$x_{313} \in \{0,1\}$

$x_{314} \in \{0,1\}$

$x_{315} \in \{0,1\}$

$x_{316} \in \{0,1\}$

$x_{321} \in \{0,1\}$

$x_{322} \in \{0,1\}$

$x_{323} \in \{0,1\}$

$x_{324} \in \{0,1\}$

$x_{325} \in \{0,1\}$

$x_{326} \in \{0,1\}$

$x_{331} \in \{0,1\}$

$x_{332} \in \{0,1\}$

$x_{333} \in \{0,1\}$

$x_{334} \in \{0,1\}$

$x_{335} \in \{0,1\}$

$x_{336} \in \{0,1\}$

$x_{341} \in \{0,1\}$

$x_{342} \in \{0,1\}$

$x_{343} \in \{0,1\}$

$x_{344} \in \{0,1\}$

$x_{345} \in \{0,1\}$

$x_{346} \in \{0,1\}$

$x_{351} \in \{0,1\}$

$x_{352} \in \{0,1\}$

$x_{353} \in \{0,1\}$

$x_{354} \in \{0,1\}$

$x_{355} \in \{0,1\}$

$x_{356} \in \{0,1\}$

$x_{361} \in \{0,1\}$

$x_{362} \in \{0,1\}$

$x_{363} \in \{0,1\}$

$x_{364} \in \{0,1\}$

$x_{365} \in \{0,1\}$

$x_{366} \in \{0,1\}$

$x_{411} \in \{0,1\}$

$x_{412} \in \{0,1\}$

$x_{413} \in \{0,1\}$

$x_{414} \in \{0,1\}$

$x_{415} \in \{0,1\}$

$x_{416} \in \{0,1\}$

$x_{421} \in \{0,1\}$

$x_{422} \in \{0,1\}$

$x_{423} \in \{0,1\}$

$x_{424} \in \{0,1\}$

$x_{425} \in \{0,1\}$

$x_{426} \in \{0,1\}$

$x_{431} \in \{0,1\}$

$x_{432} \in \{0,1\}$

$x_{433} \in \{0,1\}$

$x_{434} \in \{0,1\}$

$x_{435} \in \{0,1\}$

$x_{436} \in \{0,1\}$

$x_{441} \in \{0,1\}$

$x_{442} \in \{0,1\}$

$x_{443} \in \{0,1\}$

$x_{444} \in \{0,1\}$

$x_{445} \in \{0,1\}$

$x_{446} \in \{0,1\}$

$x_{451} \in \{0,1\}$

$x_{452} \in \{0,1\}$

$x_{453} \in \{0,1\}$

$x_{454} \in \{0,1\}$

$x_{455} \in \{0,1\}$

$x_{456} \in \{0,1\}$

$x_{461} \in \{0,1\}$

$x_{462} \in \{0,1\}$

$x_{463} \in \{0,1\}$

$x_{464} \in \{0,1\}$

$x_{465} \in \{0,1\}$

$x_{466} \in \{0,1\}$

$x_{511} \in \{0,1\}$

$x_{512} \in \{0,1\}$

$x_{513} \in \{0,1\}$

$x_{514} \in \{0,1\}$

$x_{515} \in \{0,1\}$

$x_{516} \in \{0,1\}$

$x_{521} \in \{0,1\}$

$x_{522} \in \{0,1\}$

$x_{523} \in \{0,1\}$

$x_{524} \in \{0,1\}$

$x_{525} \in \{0,1\}$

$x_{526} \in \{0,1\}$

$x_{531} \in \{0,1\}$

$x_{532} \in \{0,1\}$

$x_{533} \in \{0,1\}$

$x_{534} \in \{0,1\}$

$x_{535} \in \{0,1\}$

$x_{536} \in \{0,1\}$

$x_{541} \in \{0,1\}$

$x_{542} \in \{0,1\}$

$x_{543} \in \{0,1\}$

$x_{544} \in \{0,1\}$

$x_{545} \in \{0,1\}$

$x_{546} \in \{0,1\}$

$x_{551} \in \{0,1\}$

$x_{552} \in \{0,1\}$

$x_{553} \in \{0,1\}$

$x_{554} \in \{0,1\}$

$x_{555} \in \{0,1\}$

$x_{556} \in \{0,1\}$

$x_{561} \in \{0,1\}$

$x_{562} \in \{0,1\}$

$x_{563} \in \{0,1\}$

$x_{564} \in \{0,1\}$

$x_{565} \in \{0,1\}$

$x_{566} \in \{0,1\}$

$x_{611} \in \{0,1\}$

$x_{612} \in \{0,1\}$

$x_{613} \in \{0,1\}$

$x_{614} \in \{0,1\}$

$x_{615} \in \{0,1\}$

$x_{616} \in \{0,1\}$

$x_{621} \in \{0,1\}$

$x_{622} \in \{0,1\}$

$x_{623} \in \{0,1\}$

$x_{624} \in \{0,1\}$

$x_{625} \in \{0,1\}$

$x_{626} \in \{0,1\}$

$x_{631} \in \{0,1\}$

$x_{632} \in \{0,1\}$

$x_{633} \in \{0,1\}$

$x_{634} \in \{0,1\}$

$x_{635} \in \{0,1\}$

$x_{636} \in \{0,1\}$

$x_{641} \in \{0,1\}$

$x_{642} \in \{0,1\}$

$x_{643} \in \{0,1\}$

$x_{644} \in \{0,1\}$

$x_{645} \in \{0,1\}$

$x_{646} \in \{0,1\}$

$x_{651} \in \{0,1\}$

$x_{652} \in \{0,1\}$

$x_{653} \in \{0,1\}$

$x_{654} \in \{0,1\}$

$x_{655} \in \{0,1\}$

$x_{656} \in \{0,1\}$

$x_{661} \in \{0,1\}$

$x_{662} \in \{0,1\}$

$x_{663} \in \{0,1\}$

$x_{664} \in \{0,1\}$

$x_{665} \in \{0,1\}$

$x_{666} \in \{0,1\}$