Solving LinkedIn Mini Sudoku with Linear Optimization Rodrigo Celso de Lima Porto October 29, 2025
What is Sudoku? ¶ Figure 1: Example of a classic Sudoku puzzle. Source: Wikipedia
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 × × \times ×
How to Play Mini Sudoku ¶ Figure 2: Example of a Mini Sudoku game. Source: LinkedIn Mini Sudoku
In the case of Mini Sudoku, the game consists of a smaller board dimensions than the classic one, 6 × \times × × \times ×
Objective Fill all the empty spaces in the game grid with digits from 1 to 6. Rules Each row, column, and 2x3 blocks must be filled with a digit from 1 to 6, without repetition in each row, column or block. 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 I I J J J U U U V V V K K K
I = { 1 , ⋯ , n } I = \{1, \cdots, n\} I = { 1 , ⋯ , n } The row range, where n n n n = 6 n = 6 n = 6 J = { 1 , ⋯ , n } J = \{1, \cdots, n\} J = { 1 , ⋯ , 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 , ⋯ , n } K = \{1, \cdots, n\} K = { 1 , ⋯ , n } Range of possible values, where n n n n n n U = { 1 , ⋯ , p } U = \{1, \cdots, p\} U = { 1 , ⋯ , p } Number of rows in the game’s submatrices, where p p p V = { 1 , ⋯ , q } V = \{1, \cdots, q\} V = { 1 , ⋯ , q } Number of columns in the game’s submatrices, where q q q p q = n pq = n pq = n Sets ¶ To facilitate the definition of the constraints, it is important to define at least the set S S S F F F S v u S_{vu} S vu S 11 S_{11} S 11 S 12 S_{12} S 12 S v u S_{vu} S vu
The indices of the v v v u u u × \times × × \times ×
S = { S v u ∣ ∀ v ∈ V , ∀ u ∈ U } S = \{S_{vu} \mid \forall v \in V, \forall u \in U\} S = { S vu ∣ ∀ v ∈ V , ∀ u ∈ U } Set of submatrices S v u S_{vu} S vu S v u = { ( i , j ) ∣ ∀ i ∈ { p ( v − 1 ) + 1 , ⋯ , p v } , ∀ j ∈ { q ( u − 1 ) + 1 , ⋯ , q u } } S_{vu} = \{(i, j) \mid \forall i \in \{p(v-1)+1, \cdots, pv\}, \forall j \in \{q(u-1)+1, \cdots, qu\}\} S vu = {( i , j ) ∣ ∀ i ∈ { p ( v − 1 ) + 1 , ⋯ , p v } , ∀ j ∈ { q ( u − 1 ) + 1 , ⋯ , q u }} Set of squares ( i , j ) ⊆ I × J (i, j) \subseteq I \times J ( i , j ) ⊆ I × J S v u S_{vu} S vu F = { ( i , j , k ) ∣ i ∈ I , j ∈ J , k ∈ K } ⊆ I × J × K F = \{(i, j, k) \mid i \in I, j \in J, k \in K\} \subseteq I \times J \times K F = {( i , j , k ) ∣ i ∈ I , j ∈ J , k ∈ K } ⊆ I × J × K Subset of pre-filled squares. Decision Variables ¶ The decision variables will be binary x i j k x_{ijk} x ijk ( i , j ) (i, j) ( i , j ) k k k BLOP .
x i j k ∈ { 0 , 1 } , ∀ ( i , j , k ) ∈ I × J × K x_{ijk} \in \B, \forall (i,j,k) \in I \times J \times K x ijk ∈ { 0 , 1 } , ∀ ( i , j , k ) ∈ I × J × K x i j k = 1 x_{ijk} = 1 x ijk = 1 ( i , j ) (i,j) ( i , j ) k k k x i j k = 0 x_{ijk} = 0 x ijk = 0 Objective Function ¶ 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 i j k ∈ { 0 , 1 } , ∀ ( i , j , k ) ∈ I × J × K x_{ijk} \in \{0, 1\}, \forall (i, j, k) \in I \times J \times K x ijk ∈ { 0 , 1 } , ∀ ( i , j , k ) ∈ I × J × K Unique-Digits-Per-Row Constraints Since there can be no repetition of values for each row, the sum of the x i j k x_{ijk} x ijk i i i j j j k k k ∑ j ∈ J x i j k = 1 , ∀ i ∈ I , ∀ k ∈ K \sum_{j \in J}{x_{ijk}}=1, \forall i \in I, \forall k \in K j ∈ J ∑ x ijk = 1 , ∀ i ∈ I , ∀ k ∈ K Unique-Digits-Per-Column Constraints The same logic applies to each column j j j i i i k k k ∑ i ∈ I x i j k = 1 , ∀ j ∈ J , ∀ k ∈ K \sum_{i \in I}{x_{ijk}}=1, \forall j \in J, \forall k \in K i ∈ I ∑ x ijk = 1 , ∀ j ∈ J , ∀ k ∈ K Unique-Digits-Per-Submatrix Constraints With the set S S S S v u S_{vu} S vu ∑ ( i , j ) ∈ S v u x i j k = 1 , ∀ S v u ∈ S , ∀ k ∈ K \sum_{(i,j) \in S_{vu}}{x_{ijk}}=1, \forall S_{vu} \in S, \forall k \in K ( i , j ) ∈ S vu ∑ x ijk = 1 , ∀ S vu ∈ S , ∀ k ∈ K 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 i j k x_{ijk} x ijk ( i , j ) (i, j) ( i , j ) ∑ k ∈ K x i j k = 1 , ∀ i ∈ I , ∀ j ∈ J \sum_{k \in K}{x_{ijk}}=1, \forall i \in I, \forall j \in J k ∈ K ∑ x ijk = 1 , ∀ i ∈ I , ∀ j ∈ J Already-Filled-Squares Constraints Finally, for each already filled square, we must remember to impose that x i j k = 1 x_{ijk} = 1 x ijk = 1 ( i , j ) (i, j) ( i , j ) k k k x i j k = 1 , ∀ ( i , j , k ) ∈ F x_{ijk}=1, \forall (i,j,k) \in F x ijk = 1 , ∀ ( i , j , k ) ∈ F 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 × q p \times q p × q p q = n pq = n pq = n
S.t.: ∑ i ∈ I x i j k = 1 , ∀ j ∈ J , ∀ k ∈ K ∑ j ∈ J x i j k = 1 , ∀ i ∈ I , ∀ k ∈ K ∑ k ∈ K x i j k = 1 , ∀ i ∈ I , ∀ j ∈ J ∑ ( i , j ) ∈ S v u x i j k = 1 , ∀ S v u ∈ S , ∀ k ∈ K x i j k = 1 , ∀ ( i , j , k ) ∈ F x i j k ∈ { 0 , 1 } , ∀ ( i , j , k ) ∈ I × J × K \begin{array}{lll}
\text{S.t.:} & & \\
& \sum_{i \in I}{x_{ijk}}=1, & \forall j \in J, \forall k \in K \\
& \sum_{j \in J}{x_{ijk}}=1, & \forall i \in I, \forall k \in K \\
& \sum_{k \in K}{x_{ijk}}=1, & \forall i \in I, \forall j \in J \\
& \sum_{(i,j) \in S_{vu}}{x_{ijk}}=1, & \forall S_{vu} \in S, \forall k \in K \\
& x_{ijk}=1, & \forall (i, j,k) \in F \\
& x_{ijk} \in \B, & \forall (i,j,k) \in I \times J \times K \\
\end{array} S.t.: ∑ i ∈ I x ijk = 1 , ∑ j ∈ J x ijk = 1 , ∑ k ∈ K x ijk = 1 , ∑ ( i , j ) ∈ S vu x ijk = 1 , x ijk = 1 , x ijk ∈ { 0 , 1 } , ∀ j ∈ J , ∀ k ∈ K ∀ i ∈ I , ∀ k ∈ K ∀ i ∈ I , ∀ j ∈ J ∀ S vu ∈ S , ∀ k ∈ K ∀ ( i , j , k ) ∈ F ∀ ( i , j , k ) ∈ I × J × K Concrete Model ¶ The example to be solved in this notebook will be Mini Sudoku No. 60, published on LinkedIn on October 10th , 2025
Figure 3: Mini Sudoku No. 60, October 10th , 2025 (Source: LinkedIn MiniSudoku )
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 x_{111} + x_{121} + x_{131} + x_{141} + x_{151} + x_{161} = 1 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 x_{211} + x_{221} + x_{231} + x_{241} + x_{251} + x_{261} = 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 x_{311} + x_{321} + x_{331} + x_{341} + x_{351} + x_{361} = 1 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 x_{411} + x_{421} + x_{431} + x_{441} + x_{451} + x_{461} = 1 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 x_{511} + x_{521} + x_{531} + x_{541} + x_{551} + x_{561} = 1 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 x_{611} + x_{621} + x_{631} + x_{641} + x_{651} + x_{661} = 1 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 x_{112} + x_{122} + x_{132} + x_{142} + x_{152} + x_{162} = 1 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 x_{212} + x_{222} + x_{232} + x_{242} + x_{252} + x_{262} = 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 x_{312} + x_{322} + x_{332} + x_{342} + x_{352} + x_{362} = 1 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 x_{412} + x_{422} + x_{432} + x_{442} + x_{452} + x_{462} = 1 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 x_{512} + x_{522} + x_{532} + x_{542} + x_{552} + x_{562} = 1 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 x_{612} + x_{622} + x_{632} + x_{642} + x_{652} + x_{662} = 1 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 x_{113} + x_{123} + x_{133} + x_{143} + x_{153} + x_{163} = 1 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 x_{213} + x_{223} + x_{233} + x_{243} + x_{253} + x_{263} = 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 x_{313} + x_{323} + x_{333} + x_{343} + x_{353} + x_{363} = 1 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 x_{413} + x_{423} + x_{433} + x_{443} + x_{453} + x_{463} = 1 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 x_{513} + x_{523} + x_{533} + x_{543} + x_{553} + x_{563} = 1 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 x_{613} + x_{623} + x_{633} + x_{643} + x_{653} + x_{663} = 1 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 x_{114} + x_{124} + x_{134} + x_{144} + x_{154} + x_{164} = 1 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 x_{214} + x_{224} + x_{234} + x_{244} + x_{254} + x_{264} = 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 x_{314} + x_{324} + x_{334} + x_{344} + x_{354} + x_{364} = 1 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 x_{414} + x_{424} + x_{434} + x_{444} + x_{454} + x_{464} = 1 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 x_{514} + x_{524} + x_{534} + x_{544} + x_{554} + x_{564} = 1 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 x_{614} + x_{624} + x_{634} + x_{644} + x_{654} + x_{664} = 1 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 x_{115} + x_{125} + x_{135} + x_{145} + x_{155} + x_{165} = 1 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 x_{215} + x_{225} + x_{235} + x_{245} + x_{255} + x_{265} = 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 x_{315} + x_{325} + x_{335} + x_{345} + x_{355} + x_{365} = 1 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 x_{415} + x_{425} + x_{435} + x_{445} + x_{455} + x_{465} = 1 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 x_{515} + x_{525} + x_{535} + x_{545} + x_{555} + x_{565} = 1 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 x_{615} + x_{625} + x_{635} + x_{645} + x_{655} + x_{665} = 1 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 x_{116} + x_{126} + x_{136} + x_{146} + x_{156} + x_{166} = 1 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 x_{216} + x_{226} + x_{236} + x_{246} + x_{256} + x_{266} = 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 x_{316} + x_{326} + x_{336} + x_{346} + x_{356} + x_{366} = 1 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 x_{416} + x_{426} + x_{436} + x_{446} + x_{456} + x_{466} = 1 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 x_{516} + x_{526} + x_{536} + x_{546} + x_{556} + x_{566} = 1 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 x_{616} + x_{626} + x_{636} + x_{646} + x_{656} + x_{666} = 1 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 x_{111} + x_{211} + x_{311} + x_{411} + x_{511} + x_{611} = 1 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 x_{121} + x_{221} + x_{321} + x_{421} + x_{521} + x_{621} = 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 x_{131} + x_{231} + x_{331} + x_{431} + x_{531} + x_{631} = 1 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 x_{141} + x_{241} + x_{341} + x_{441} + x_{541} + x_{641} = 1 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 x_{151} + x_{251} + x_{351} + x_{451} + x_{551} + x_{651} = 1 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 x_{161} + x_{261} + x_{361} + x_{461} + x_{561} + x_{661} = 1 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 x_{112} + x_{212} + x_{312} + x_{412} + x_{512} + x_{612} = 1 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 x_{122} + x_{222} + x_{322} + x_{422} + x_{522} + x_{622} = 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 x_{132} + x_{232} + x_{332} + x_{432} + x_{532} + x_{632} = 1 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 x_{142} + x_{242} + x_{342} + x_{442} + x_{542} + x_{642} = 1 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 x_{152} + x_{252} + x_{352} + x_{452} + x_{552} + x_{652} = 1 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 x_{162} + x_{262} + x_{362} + x_{462} + x_{562} + x_{662} = 1 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 x_{113} + x_{213} + x_{313} + x_{413} + x_{513} + x_{613} = 1 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 x_{123} + x_{223} + x_{323} + x_{423} + x_{523} + x_{623} = 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 x_{133} + x_{233} + x_{333} + x_{433} + x_{533} + x_{633} = 1 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 x_{143} + x_{243} + x_{343} + x_{443} + x_{543} + x_{643} = 1 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 x_{153} + x_{253} + x_{353} + x_{453} + x_{553} + x_{653} = 1 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 x_{163} + x_{263} + x_{363} + x_{463} + x_{563} + x_{663} = 1 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 x_{114} + x_{214} + x_{314} + x_{414} + x_{514} + x_{614} = 1 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 x_{124} + x_{224} + x_{324} + x_{424} + x_{524} + x_{624} = 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 x_{134} + x_{234} + x_{334} + x_{434} + x_{534} + x_{634} = 1 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 x_{144} + x_{244} + x_{344} + x_{444} + x_{544} + x_{644} = 1 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 x_{154} + x_{254} + x_{354} + x_{454} + x_{554} + x_{654} = 1 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 x_{164} + x_{264} + x_{364} + x_{464} + x_{564} + x_{664} = 1 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 x_{115} + x_{215} + x_{315} + x_{415} + x_{515} + x_{615} = 1 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 x_{125} + x_{225} + x_{325} + x_{425} + x_{525} + x_{625} = 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 x_{135} + x_{235} + x_{335} + x_{435} + x_{535} + x_{635} = 1 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 x_{145} + x_{245} + x_{345} + x_{445} + x_{545} + x_{645} = 1 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 x_{155} + x_{255} + x_{355} + x_{455} + x_{555} + x_{655} = 1 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 x_{165} + x_{265} + x_{365} + x_{465} + x_{565} + x_{665} = 1 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 x_{116} + x_{216} + x_{316} + x_{416} + x_{516} + x_{616} = 1 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 x_{126} + x_{226} + x_{326} + x_{426} + x_{526} + x_{626} = 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 x_{136} + x_{236} + x_{336} + x_{436} + x_{536} + x_{636} = 1 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 x_{146} + x_{246} + x_{346} + x_{446} + x_{546} + x_{646} = 1 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 x_{156} + x_{256} + x_{356} + x_{456} + x_{556} + x_{656} = 1 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 x_{166} + x_{266} + x_{366} + x_{466} + x_{566} + x_{666} = 1 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 x_{111} + x_{121} + x_{131} + x_{211} + x_{221} + x_{231} = 1 x 111 + x 121 + x 131 + x 211 + x 221 + x 231 = 1 (Digit 1 on Submatrix S 11 S_{11} S 11 x 112 + x 122 + x 132 + x 212 + x 222 + x 232 = 1 x_{112} + x_{122} + x_{132} + x_{212} + x_{222} + x_{232} = 1 x 112 + x 122 + x 132 + x 212 + x 222 + x 232 = 1 (Digit 2 on Submatrix S 11 S_{11} S 11 x 113 + x 123 + x 133 + x 213 + x 223 + x 233 = 1 x_{113} + x_{123} + x_{133} + x_{213} + x_{223} + x_{233} = 1 x 113 + x 123 + x 133 + x 213 + x 223 + x 233 = 1 (Digit 3 on Submatrix S 11 S_{11} S 11 x 114 + x 124 + x 134 + x 214 + x 224 + x 234 = 1 x_{114} + x_{124} + x_{134} + x_{214} + x_{224} + x_{234} = 1 x 114 + x 124 + x 134 + x 214 + x 224 + x 234 = 1 (Digit 4 on Submatrix S 11 S_{11} S 11 x 115 + x 125 + x 135 + x 215 + x 225 + x 235 = 1 x_{115} + x_{125} + x_{135} + x_{215} + x_{225} + x_{235} = 1 x 115 + x 125 + x 135 + x 215 + x 225 + x 235 = 1 (Digit 5 on Submatrix S 11 S_{11} S 11 x 116 + x 126 + x 136 + x 216 + x 226 + x 236 = 1 x_{116} + x_{126} + x_{136} + x_{216} + x_{226} + x_{236} = 1 x 116 + x 126 + x 136 + x 216 + x 226 + x 236 = 1 (Digit 6 on Submatrix S 11 S_{11} S 11 x 141 + x 151 + x 161 + x 241 + x 251 + x 261 = 1 x_{141} + x_{151} + x_{161} + x_{241} + x_{251} + x_{261} = 1 x 141 + x 151 + x 161 + x 241 + x 251 + x 261 = 1 (Digit 1 on Submatrix S 12 S_{12} S 12 x 142 + x 152 + x 162 + x 242 + x 252 + x 262 = 1 x_{142} + x_{152} + x_{162} + x_{242} + x_{252} + x_{262} = 1 x 142 + x 152 + x 162 + x 242 + x 252 + x 262 = 1 (Digit 2 on Submatrix S 12 S_{12} S 12 x 143 + x 153 + x 163 + x 243 + x 253 + x 263 = 1 x_{143} + x_{153} + x_{163} + x_{243} + x_{253} + x_{263} = 1 x 143 + x 153 + x 163 + x 243 + x 253 + x 263 = 1 (Digit 3 on Submatrix S 12 S_{12} S 12 x 144 + x 154 + x 164 + x 244 + x 254 + x 264 = 1 x_{144} + x_{154} + x_{164} + x_{244} + x_{254} + x_{264} = 1 x 144 + x 154 + x 164 + x 244 + x 254 + x 264 = 1 (Digit 4 on Submatrix S 12 S_{12} S 12 x 145 + x 155 + x 165 + x 245 + x 255 + x 265 = 1 x_{145} + x_{155} + x_{165} + x_{245} + x_{255} + x_{265} = 1 x 145 + x 155 + x 165 + x 245 + x 255 + x 265 = 1 (Digit 5 on Submatrix S 12 S_{12} S 12 x 146 + x 156 + x 166 + x 246 + x 256 + x 266 = 1 x_{146} + x_{156} + x_{166} + x_{246} + x_{256} + x_{266} = 1 x 146 + x 156 + x 166 + x 246 + x 256 + x 266 = 1 (Digit 6 on Submatrix S 12 S_{12} S 12 x 311 + x 321 + x 331 + x 411 + x 421 + x 431 = 1 x_{311} + x_{321} + x_{331} + x_{411} + x_{421} + x_{431} = 1 x 311 + x 321 + x 331 + x 411 + x 421 + x 431 = 1 (Digit 1 on Submatrix S 21 S_{21} S 21 x 312 + x 322 + x 332 + x 412 + x 422 + x 432 = 1 x_{312} + x_{322} + x_{332} + x_{412} + x_{422} + x_{432} = 1 x 312 + x 322 + x 332 + x 412 + x 422 + x 432 = 1 (Digit 2 on Submatrix S 21 S_{21} S 21 x 313 + x 323 + x 333 + x 413 + x 423 + x 433 = 1 x_{313} + x_{323} + x_{333} + x_{413} + x_{423} + x_{433} = 1 x 313 + x 323 + x 333 + x 413 + x 423 + x 433 = 1 (Digit 3 on Submatrix S 21 S_{21} S 21 x 314 + x 324 + x 334 + x 414 + x 424 + x 434 = 1 x_{314} + x_{324} + x_{334} + x_{414} + x_{424} + x_{434} = 1 x 314 + x 324 + x 334 + x 414 + x 424 + x 434 = 1 (Digit 4 on Submatrix S 21 S_{21} S 21 x 315 + x 325 + x 335 + x 415 + x 425 + x 435 = 1 x_{315} + x_{325} + x_{335} + x_{415} + x_{425} + x_{435} = 1 x 315 + x 325 + x 335 + x 415 + x 425 + x 435 = 1 (Digit 5 on Submatrix S 21 S_{21} S 21 x 316 + x 326 + x 336 + x 416 + x 426 + x 436 = 1 x_{316} + x_{326} + x_{336} + x_{416} + x_{426} + x_{436} = 1 x 316 + x 326 + x 336 + x 416 + x 426 + x 436 = 1 (Digit 6 on Submatrix S 21 S_{21} S 21 x 341 + x 351 + x 331 + x 441 + x 451 + x 431 = 1 x_{341} + x_{351} + x_{331} + x_{441} + x_{451} + x_{431} = 1 x 341 + x 351 + x 331 + x 441 + x 451 + x 431 = 1 (Digit 1 on Submatrix S 22 S_{22} S 22 x 342 + x 352 + x 362 + x 442 + x 452 + x 462 = 1 x_{342} + x_{352} + x_{362} + x_{442} + x_{452} + x_{462} = 1 x 342 + x 352 + x 362 + x 442 + x 452 + x 462 = 1 (Digit 2 on Submatrix S 22 S_{22} S 22 x 343 + x 353 + x 363 + x 443 + x 453 + x 463 = 1 x_{343} + x_{353} + x_{363} + x_{443} + x_{453} + x_{463} = 1 x 343 + x 353 + x 363 + x 443 + x 453 + x 463 = 1 (Digit 3 on Submatrix S 22 S_{22} S 22 x 344 + x 354 + x 364 + x 444 + x 454 + x 464 = 1 x_{344} + x_{354} + x_{364} + x_{444} + x_{454} + x_{464} = 1 x 344 + x 354 + x 364 + x 444 + x 454 + x 464 = 1 (Digit 4 on Submatrix S 22 S_{22} S 22 x 345 + x 355 + x 365 + x 445 + x 455 + x 465 = 1 x_{345} + x_{355} + x_{365} + x_{445} + x_{455} + x_{465} = 1 x 345 + x 355 + x 365 + x 445 + x 455 + x 465 = 1 (Digit 5 on Submatrix S 22 S_{22} S 22 x 346 + x 356 + x 366 + x 446 + x 456 + x 466 = 1 x_{346} + x_{356} + x_{366} + x_{446} + x_{456} + x_{466} = 1 x 346 + x 356 + x 366 + x 446 + x 456 + x 466 = 1 (Digit 6 on Submatrix S 22 S_{22} S 22 x 511 + x 521 + x 531 + x 611 + x 621 + x 631 = 1 x_{511} + x_{521} + x_{531} + x_{611} + x_{621} + x_{631} = 1 x 511 + x 521 + x 531 + x 611 + x 621 + x 631 = 1 (Digit 1 on Submatrix S 31 S_{31} S 31 x 512 + x 522 + x 532 + x 612 + x 622 + x 632 = 1 x_{512} + x_{522} + x_{532} + x_{612} + x_{622} + x_{632} = 1 x 512 + x 522 + x 532 + x 612 + x 622 + x 632 = 1 (Digit 2 on Submatrix S 31 S_{31} S 31 x 513 + x 523 + x 533 + x 613 + x 623 + x 633 = 1 x_{513} + x_{523} + x_{533} + x_{613} + x_{623} + x_{633} = 1 x 513 + x 523 + x 533 + x 613 + x 623 + x 633 = 1 (Digit 3 on Submatrix S 31 S_{31} S 31 x 514 + x 524 + x 534 + x 614 + x 624 + x 634 = 1 x_{514} + x_{524} + x_{534} + x_{614} + x_{624} + x_{634} = 1 x 514 + x 524 + x 534 + x 614 + x 624 + x 634 = 1 (Digit 4 on Submatrix S 31 S_{31} S 31 x 515 + x 525 + x 535 + x 615 + x 625 + x 635 = 1 x_{515} + x_{525} + x_{535} + x_{615} + x_{625} + x_{635} = 1 x 515 + x 525 + x 535 + x 615 + x 625 + x 635 = 1 (Digit 5 on Submatrix S 31 S_{31} S 31 x 516 + x 526 + x 536 + x 616 + x 626 + x 636 = 1 x_{516} + x_{526} + x_{536} + x_{616} + x_{626} + x_{636} = 1 x 516 + x 526 + x 536 + x 616 + x 626 + x 636 = 1 (Digit 6 on Submatrix S 31 S_{31} S 31 x 541 + x 551 + x 561 + x 641 + x 651 + x 631 = 1 x_{541} + x_{551} + x_{561} + x_{641} + x_{651} + x_{631} = 1 x 541 + x 551 + x 561 + x 641 + x 651 + x 631 = 1 (Digit 1 on Submatrix S 32 S_{32} S 32 x 542 + x 552 + x 562 + x 642 + x 652 + x 662 = 1 x_{542} + x_{552} + x_{562} + x_{642} + x_{652} + x_{662} = 1 x 542 + x 552 + x 562 + x 642 + x 652 + x 662 = 1 (Digit 2 on Submatrix S 32 S_{32} S 32 x 543 + x 553 + x 563 + x 643 + x 653 + x 663 = 1 x_{543} + x_{553} + x_{563} + x_{643} + x_{653} + x_{663} = 1 x 543 + x 553 + x 563 + x 643 + x 653 + x 663 = 1 (Digit 3 on Submatrix S 32 S_{32} S 32 x 544 + x 554 + x 564 + x 644 + x 654 + x 664 = 1 x_{544} + x_{554} + x_{564} + x_{644} + x_{654} + x_{664} = 1 x 544 + x 554 + x 564 + x 644 + x 654 + x 664 = 1 (Digit 4 on Submatrix S 32 S_{32} S 32 x 545 + x 555 + x 565 + x 645 + x 655 + x 665 = 1 x_{545} + x_{555} + x_{565} + x_{645} + x_{655} + x_{665} = 1 x 545 + x 555 + x 565 + x 645 + x 655 + x 665 = 1 (Digit 5 on Submatrix S 32 S_{32} S 32 x 546 + x 556 + x 566 + x 646 + x 656 + x 666 = 1 x_{546} + x_{556} + x_{566} + x_{646} + x_{656} + x_{666} = 1 x 546 + x 556 + x 566 + x 646 + x 656 + x 666 = 1 (Digit 6 on Submatrix S 32 S_{32} S 32 Single-Digit-Per-Square Constraints x 111 + x 112 + x 113 + x 114 + x 115 + x 116 = 1 x_{111} + x_{112} + x_{113} + x_{114} + x_{115} + x_{116} = 1 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 x_{121} + x_{122} + x_{123} + x_{124} + x_{125} + x_{126} = 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 x_{131} + x_{132} + x_{133} + x_{134} + x_{135} + x_{136} = 1 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 x_{141} + x_{142} + x_{143} + x_{144} + x_{145} + x_{146} = 1 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 x_{151} + x_{152} + x_{153} + x_{154} + x_{155} + x_{156} = 1 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 x_{161} + x_{162} + x_{163} + x_{164} + x_{165} + x_{166} = 1 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 x_{211} + x_{212} + x_{213} + x_{214} + x_{215} + x_{216} = 1 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 x_{221} + x_{222} + x_{223} + x_{224} + x_{225} + x_{226} = 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 x_{231} + x_{232} + x_{233} + x_{234} + x_{235} + x_{236} = 1 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 x_{241} + x_{242} + x_{243} + x_{244} + x_{245} + x_{246} = 1 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 x_{251} + x_{252} + x_{253} + x_{254} + x_{255} + x_{256} = 1 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 x_{261} + x_{262} + x_{263} + x_{264} + x_{265} + x_{266} = 1 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 x_{311} + x_{312} + x_{313} + x_{314} + x_{315} + x_{316} = 1 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 x_{321} + x_{322} + x_{323} + x_{324} + x_{325} + x_{326} = 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 x_{331} + x_{332} + x_{333} + x_{334} + x_{335} + x_{336} = 1 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 x_{341} + x_{342} + x_{343} + x_{344} + x_{345} + x_{346} = 1 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 x_{351} + x_{352} + x_{353} + x_{354} + x_{355} + x_{356} = 1 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 x_{361} + x_{362} + x_{363} + x_{364} + x_{365} + x_{366} = 1 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 x_{411} + x_{412} + x_{413} + x_{414} + x_{415} + x_{416} = 1 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 x_{421} + x_{422} + x_{423} + x_{424} + x_{425} + x_{426} = 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 x_{431} + x_{432} + x_{433} + x_{434} + x_{435} + x_{436} = 1 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 x_{441} + x_{442} + x_{443} + x_{444} + x_{445} + x_{446} = 1 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 x_{451} + x_{452} + x_{453} + x_{454} + x_{455} + x_{456} = 1 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 x_{461} + x_{462} + x_{463} + x_{464} + x_{465} + x_{466} = 1 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 x_{511} + x_{512} + x_{513} + x_{514} + x_{515} + x_{516} = 1 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 x_{521} + x_{522} + x_{523} + x_{524} + x_{525} + x_{526} = 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 x_{531} + x_{532} + x_{533} + x_{534} + x_{535} + x_{536} = 1 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 x_{541} + x_{542} + x_{543} + x_{544} + x_{545} + x_{546} = 1 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 x_{551} + x_{552} + x_{553} + x_{554} + x_{555} + x_{556} = 1 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 x_{561} + x_{562} + x_{563} + x_{564} + x_{565} + x_{566} = 1 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 x_{611} + x_{612} + x_{613} + x_{614} + x_{615} + x_{616} = 1 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 x_{621} + x_{622} + x_{623} + x_{624} + x_{625} + x_{626} = 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 x_{631} + x_{632} + x_{633} + x_{634} + x_{635} + x_{636} = 1 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 x_{641} + x_{642} + x_{643} + x_{644} + x_{645} + x_{646} = 1 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 x_{651} + x_{652} + x_{653} + x_{654} + x_{655} + x_{656} = 1 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 x_{661} + x_{662} + x_{663} + x_{664} + x_{665} + x_{666} = 1 x 661 + x 662 + x 663 + x 664 + x 665 + x 666 = 1 (Square (6, 6) ) Already-Filled-Squares Constraints x 111 = 1 x_{111} = 1 x 111 = 1 (Square (1, 1) has Digit 1 ) x 222 = 1 x_{222} = 1 x 222 = 1 (Square (2, 2) has Digit 2 ) x 253 = 1 x_{253} = 1 x 253 = 1 (Square (2, 5) has Digit 3 ) x 346 = 1 x_{346} = 1 x 346 = 1 (Square (3, 4) has Digit 6 ) x 435 = 1 x_{435} = 1 x 435 = 1 (Square (4, 3) has Digit 5 ) x 444 = 1 x_{444} = 1 x 444 = 1 (Square (4, 4) has Digit 4 ) x 524 = 1 x_{524} = 1 x 524 = 1 (Square (5, 2) has Digit 4 ) x 555 = 1 x_{555} = 1 x 555 = 1 (Square (5, 5) has Digit 5 ) x 666 = 1 x_{666} = 1 x 666 = 1 (Square (6, 6) has Digit 6 ) Binary Constraints x 111 ∈ { 0 , 1 } x_{111} \in \B x 111 ∈ { 0 , 1 } x 112 ∈ { 0 , 1 } x_{112} \in \B x 112 ∈ { 0 , 1 } x 113 ∈ { 0 , 1 } x_{113} \in \B x 113 ∈ { 0 , 1 } x 114 ∈ { 0 , 1 } x_{114} \in \B x 114 ∈ { 0 , 1 } x 115 ∈ { 0 , 1 } x_{115} \in \B x 115 ∈ { 0 , 1 } x 116 ∈ { 0 , 1 } x_{116} \in \B x 116 ∈ { 0 , 1 } x 121 ∈ { 0 , 1 } x_{121} \in \B x 121 ∈ { 0 , 1 } x 122 ∈ { 0 , 1 } x_{122} \in \B x 122 ∈ { 0 , 1 } x 123 ∈ { 0 , 1 } x_{123} \in \B x 123 ∈ { 0 , 1 } x 124 ∈ { 0 , 1 } x_{124} \in \B x 124 ∈ { 0 , 1 } x 125 ∈ { 0 , 1 } x_{125} \in \B x 125 ∈ { 0 , 1 } x 126 ∈ { 0 , 1 } x_{126} \in \B x 126 ∈ { 0 , 1 } x 131 ∈ { 0 , 1 } x_{131} \in \B x 131 ∈ { 0 , 1 } x 132 ∈ { 0 , 1 } x_{132} \in \B x 132 ∈ { 0 , 1 } x 133 ∈ { 0 , 1 } x_{133} \in \B x 133 ∈ { 0 , 1 } x 134 ∈ { 0 , 1 } x_{134} \in \B x 134 ∈ { 0 , 1 } x 135 ∈ { 0 , 1 } x_{135} \in \B x 135 ∈ { 0 , 1 } x 136 ∈ { 0 , 1 } x_{136} \in \B x 136 ∈ { 0 , 1 } x 141 ∈ { 0 , 1 } x_{141} \in \B x 141 ∈ { 0 , 1 } x 142 ∈ { 0 , 1 } x_{142} \in \B x 142 ∈ { 0 , 1 } x 143 ∈ { 0 , 1 } x_{143} \in \B x 143 ∈ { 0 , 1 } x 144 ∈ { 0 , 1 } x_{144} \in \B x 144 ∈ { 0 , 1 } x 145 ∈ { 0 , 1 } x_{145} \in \B x 145 ∈ { 0 , 1 } x 146 ∈ { 0 , 1 } x_{146} \in \B x 146 ∈ { 0 , 1 } x 151 ∈ { 0 , 1 } x_{151} \in \B x 151 ∈ { 0 , 1 } x 152 ∈ { 0 , 1 } x_{152} \in \B x 152 ∈ { 0 , 1 } x 153 ∈ { 0 , 1 } x_{153} \in \B x 153 ∈ { 0 , 1 } x 154 ∈ { 0 , 1 } x_{154} \in \B x 154 ∈ { 0 , 1 } x 155 ∈ { 0 , 1 } x_{155} \in \B x 155 ∈ { 0 , 1 } x 156 ∈ { 0 , 1 } x_{156} \in \B x 156 ∈ { 0 , 1 } x 161 ∈ { 0 , 1 } x_{161} \in \B x 161 ∈ { 0 , 1 } x 162 ∈ { 0 , 1 } x_{162} \in \B x 162 ∈ { 0 , 1 } x 163 ∈ { 0 , 1 } x_{163} \in \B x 163 ∈ { 0 , 1 } x 164 ∈ { 0 , 1 } x_{164} \in \B x 164 ∈ { 0 , 1 } x 165 ∈ { 0 , 1 } x_{165} \in \B x 165 ∈ { 0 , 1 } x 166 ∈ { 0 , 1 } x_{166} \in \B x 166 ∈ { 0 , 1 } x 211 ∈ { 0 , 1 } x_{211} \in \B x 211 ∈ { 0 , 1 } x 212 ∈ { 0 , 1 } x_{212} \in \B x 212 ∈ { 0 , 1 } x 213 ∈ { 0 , 1 } x_{213} \in \B x 213 ∈ { 0 , 1 } x 214 ∈ { 0 , 1 } x_{214} \in \B x 214 ∈ { 0 , 1 } x 215 ∈ { 0 , 1 } x_{215} \in \B x 215 ∈ { 0 , 1 } x 216 ∈ { 0 , 1 } x_{216} \in \B x 216 ∈ { 0 , 1 } x 221 ∈ { 0 , 1 } x_{221} \in \B x 221 ∈ { 0 , 1 } x 222 ∈ { 0 , 1 } x_{222} \in \B x 222 ∈ { 0 , 1 } x 223 ∈ { 0 , 1 } x_{223} \in \B x 223 ∈ { 0 , 1 } x 224 ∈ { 0 , 1 } x_{224} \in \B x 224 ∈ { 0 , 1 } x 225 ∈ { 0 , 1 } x_{225} \in \B x 225 ∈ { 0 , 1 } x 226 ∈ { 0 , 1 } x_{226} \in \B x 226 ∈ { 0 , 1 } x 231 ∈ { 0 , 1 } x_{231} \in \B x 231 ∈ { 0 , 1 } x 232 ∈ { 0 , 1 } x_{232} \in \B x 232 ∈ { 0 , 1 } x 233 ∈ { 0 , 1 } x_{233} \in \B x 233 ∈ { 0 , 1 } x 234 ∈ { 0 , 1 } x_{234} \in \B x 234 ∈ { 0 , 1 } x 235 ∈ { 0 , 1 } x_{235} \in \B x 235 ∈ { 0 , 1 } x 236 ∈ { 0 , 1 } x_{236} \in \B x 236 ∈ { 0 , 1 } x 241 ∈ { 0 , 1 } x_{241} \in \B x 241 ∈ { 0 , 1 } x 242 ∈ { 0 , 1 } x_{242} \in \B x 242 ∈ { 0 , 1 } x 243 ∈ { 0 , 1 } x_{243} \in \B x 243 ∈ { 0 , 1 } x 244 ∈ { 0 , 1 } x_{244} \in \B x 244 ∈ { 0 , 1 } x 245 ∈ { 0 , 1 } x_{245} \in \B x 245 ∈ { 0 , 1 } x 246 ∈ { 0 , 1 } x_{246} \in \B x 246 ∈ { 0 , 1 } x 251 ∈ { 0 , 1 } x_{251} \in \B x 251 ∈ { 0 , 1 } x 252 ∈ { 0 , 1 } x_{252} \in \B x 252 ∈ { 0 , 1 } x 253 ∈ { 0 , 1 } x_{253} \in \B x 253 ∈ { 0 , 1 } x 254 ∈ { 0 , 1 } x_{254} \in \B x 254 ∈ { 0 , 1 } x 255 ∈ { 0 , 1 } x_{255} \in \B x 255 ∈ { 0 , 1 } x 256 ∈ { 0 , 1 } x_{256} \in \B x 256 ∈ { 0 , 1 } x 261 ∈ { 0 , 1 } x_{261} \in \B x 261 ∈ { 0 , 1 } x 262 ∈ { 0 , 1 } x_{262} \in \B x 262 ∈ { 0 , 1 } x 263 ∈ { 0 , 1 } x_{263} \in \B x 263 ∈ { 0 , 1 } x 264 ∈ { 0 , 1 } x_{264} \in \B x 264 ∈ { 0 , 1 } x 265 ∈ { 0 , 1 } x_{265} \in \B x 265 ∈ { 0 , 1 } x 266 ∈ { 0 , 1 } x_{266} \in \B x 266 ∈ { 0 , 1 } x 311 ∈ { 0 , 1 } x_{311} \in \B x 311 ∈ { 0 , 1 } x 312 ∈ { 0 , 1 } x_{312} \in \B x 312 ∈ { 0 , 1 } x 313 ∈ { 0 , 1 } x_{313} \in \B x 313 ∈ { 0 , 1 } x 314 ∈ { 0 , 1 } x_{314} \in \B x 314 ∈ { 0 , 1 } x 315 ∈ { 0 , 1 } x_{315} \in \B x 315 ∈ { 0 , 1 } x 316 ∈ { 0 , 1 } x_{316} \in \B x 316 ∈ { 0 , 1 } x 321 ∈ { 0 , 1 } x_{321} \in \B x 321 ∈ { 0 , 1 } x 322 ∈ { 0 , 1 } x_{322} \in \B x 322 ∈ { 0 , 1 } x 323 ∈ { 0 , 1 } x_{323} \in \B x 323 ∈ { 0 , 1 } x 324 ∈ { 0 , 1 } x_{324} \in \B x 324 ∈ { 0 , 1 } x 325 ∈ { 0 , 1 } x_{325} \in \B x 325 ∈ { 0 , 1 } x 326 ∈ { 0 , 1 } x_{326} \in \B x 326 ∈ { 0 , 1 } x 331 ∈ { 0 , 1 } x_{331} \in \B x 331 ∈ { 0 , 1 } x 332 ∈ { 0 , 1 } x_{332} \in \B x 332 ∈ { 0 , 1 } x 333 ∈ { 0 , 1 } x_{333} \in \B x 333 ∈ { 0 , 1 } x 334 ∈ { 0 , 1 } x_{334} \in \B x 334 ∈ { 0 , 1 } x 335 ∈ { 0 , 1 } x_{335} \in \B x 335 ∈ { 0 , 1 } x 336 ∈ { 0 , 1 } x_{336} \in \B x 336 ∈ { 0 , 1 } x 341 ∈ { 0 , 1 } x_{341} \in \B x 341 ∈ { 0 , 1 } x 342 ∈ { 0 , 1 } x_{342} \in \B x 342 ∈ { 0 , 1 } x 343 ∈ { 0 , 1 } x_{343} \in \B x 343 ∈ { 0 , 1 } x 344 ∈ { 0 , 1 } x_{344} \in \B x 344 ∈ { 0 , 1 } x 345 ∈ { 0 , 1 } x_{345} \in \B x 345 ∈ { 0 , 1 } x 346 ∈ { 0 , 1 } x_{346} \in \B x 346 ∈ { 0 , 1 } x 351 ∈ { 0 , 1 } x_{351} \in \B x 351 ∈ { 0 , 1 } x 352 ∈ { 0 , 1 } x_{352} \in \B x 352 ∈ { 0 , 1 } x 353 ∈ { 0 , 1 } x_{353} \in \B x 353 ∈ { 0 , 1 } x 354 ∈ { 0 , 1 } x_{354} \in \B x 354 ∈ { 0 , 1 } x 355 ∈ { 0 , 1 } x_{355} \in \B x 355 ∈ { 0 , 1 } x 356 ∈ { 0 , 1 } x_{356} \in \B x 356 ∈ { 0 , 1 } x 361 ∈ { 0 , 1 } x_{361} \in \B x 361 ∈ { 0 , 1 } x 362 ∈ { 0 , 1 } x_{362} \in \B x 362 ∈ { 0 , 1 } x 363 ∈ { 0 , 1 } x_{363} \in \B x 363 ∈ { 0 , 1 } x 364 ∈ { 0 , 1 } x_{364} \in \B x 364 ∈ { 0 , 1 } x 365 ∈ { 0 , 1 } x_{365} \in \B x 365 ∈ { 0 , 1 } x 366 ∈ { 0 , 1 } x_{366} \in \B x 366 ∈ { 0 , 1 } x 411 ∈ { 0 , 1 } x_{411} \in \B x 411 ∈ { 0 , 1 } x 412 ∈ { 0 , 1 } x_{412} \in \B x 412 ∈ { 0 , 1 } x 413 ∈ { 0 , 1 } x_{413} \in \B x 413 ∈ { 0 , 1 } x 414 ∈ { 0 , 1 } x_{414} \in \B x 414 ∈ { 0 , 1 } x 415 ∈ { 0 , 1 } x_{415} \in \B x 415 ∈ { 0 , 1 } x 416 ∈ { 0 , 1 } x_{416} \in \B x 416 ∈ { 0 , 1 } x 421 ∈ { 0 , 1 } x_{421} \in \B x 421 ∈ { 0 , 1 } x 422 ∈ { 0 , 1 } x_{422} \in \B x 422 ∈ { 0 , 1 } x 423 ∈ { 0 , 1 } x_{423} \in \B x 423 ∈ { 0 , 1 } x 424 ∈ { 0 , 1 } x_{424} \in \B x 424 ∈ { 0 , 1 } x 425 ∈ { 0 , 1 } x_{425} \in \B x 425 ∈ { 0 , 1 } x 426 ∈ { 0 , 1 } x_{426} \in \B x 426 ∈ { 0 , 1 } x 431 ∈ { 0 , 1 } x_{431} \in \B x 431 ∈ { 0 , 1 } x 432 ∈ { 0 , 1 } x_{432} \in \B x 432 ∈ { 0 , 1 } x 433 ∈ { 0 , 1 } x_{433} \in \B x 433 ∈ { 0 , 1 } x 434 ∈ { 0 , 1 } x_{434} \in \B x 434 ∈ { 0 , 1 } x 435 ∈ { 0 , 1 } x_{435} \in \B x 435 ∈ { 0 , 1 } x 436 ∈ { 0 , 1 } x_{436} \in \B x 436 ∈ { 0 , 1 } x 441 ∈ { 0 , 1 } x_{441} \in \B x 441 ∈ { 0 , 1 } x 442 ∈ { 0 , 1 } x_{442} \in \B x 442 ∈ { 0 , 1 } x 443 ∈ { 0 , 1 } x_{443} \in \B x 443 ∈ { 0 , 1 } x 444 ∈ { 0 , 1 } x_{444} \in \B x 444 ∈ { 0 , 1 } x 445 ∈ { 0 , 1 } x_{445} \in \B x 445 ∈ { 0 , 1 } x 446 ∈ { 0 , 1 } x_{446} \in \B x 446 ∈ { 0 , 1 } x 451 ∈ { 0 , 1 } x_{451} \in \B x 451 ∈ { 0 , 1 } x 452 ∈ { 0 , 1 } x_{452} \in \B x 452 ∈ { 0 , 1 } x 453 ∈ { 0 , 1 } x_{453} \in \B x 453 ∈ { 0 , 1 } x 454 ∈ { 0 , 1 } x_{454} \in \B x 454 ∈ { 0 , 1 } x 455 ∈ { 0 , 1 } x_{455} \in \B x 455 ∈ { 0 , 1 } x 456 ∈ { 0 , 1 } x_{456} \in \B x 456 ∈ { 0 , 1 } x 461 ∈ { 0 , 1 } x_{461} \in \B x 461 ∈ { 0 , 1 } x 462 ∈ { 0 , 1 } x_{462} \in \B x 462 ∈ { 0 , 1 } x 463 ∈ { 0 , 1 } x_{463} \in \B x 463 ∈ { 0 , 1 } x 464 ∈ { 0 , 1 } x_{464} \in \B x 464 ∈ { 0 , 1 } x 465 ∈ { 0 , 1 } x_{465} \in \B x 465 ∈ { 0 , 1 } x 466 ∈ { 0 , 1 } x_{466} \in \B x 466 ∈ { 0 , 1 } x 511 ∈ { 0 , 1 } x_{511} \in \B x 511 ∈ { 0 , 1 } x 512 ∈ { 0 , 1 } x_{512} \in \B x 512 ∈ { 0 , 1 } x 513 ∈ { 0 , 1 } x_{513} \in \B x 513 ∈ { 0 , 1 } x 514 ∈ { 0 , 1 } x_{514} \in \B x 514 ∈ { 0 , 1 } x 515 ∈ { 0 , 1 } x_{515} \in \B x 515 ∈ { 0 , 1 } x 516 ∈ { 0 , 1 } x_{516} \in \B x 516 ∈ { 0 , 1 } x 521 ∈ { 0 , 1 } x_{521} \in \B x 521 ∈ { 0 , 1 } x 522 ∈ { 0 , 1 } x_{522} \in \B x 522 ∈ { 0 , 1 } x 523 ∈ { 0 , 1 } x_{523} \in \B x 523 ∈ { 0 , 1 } x 524 ∈ { 0 , 1 } x_{524} \in \B x 524 ∈ { 0 , 1 } x 525 ∈ { 0 , 1 } x_{525} \in \B x 525 ∈ { 0 , 1 } x 526 ∈ { 0 , 1 } x_{526} \in \B x 526 ∈ { 0 , 1 } x 531 ∈ { 0 , 1 } x_{531} \in \B x 531 ∈ { 0 , 1 } x 532 ∈ { 0 , 1 } x_{532} \in \B x 532 ∈ { 0 , 1 } x 533 ∈ { 0 , 1 } x_{533} \in \B x 533 ∈ { 0 , 1 } x 534 ∈ { 0 , 1 } x_{534} \in \B x 534 ∈ { 0 , 1 } x 535 ∈ { 0 , 1 } x_{535} \in \B x 535 ∈ { 0 , 1 } x 536 ∈ { 0 , 1 } x_{536} \in \B x 536 ∈ { 0 , 1 } x 541 ∈ { 0 , 1 } x_{541} \in \B x 541 ∈ { 0 , 1 } x 542 ∈ { 0 , 1 } x_{542} \in \B x 542 ∈ { 0 , 1 } x 543 ∈ { 0 , 1 } x_{543} \in \B x 543 ∈ { 0 , 1 } x 544 ∈ { 0 , 1 } x_{544} \in \B x 544 ∈ { 0 , 1 } x 545 ∈ { 0 , 1 } x_{545} \in \B x 545 ∈ { 0 , 1 } x 546 ∈ { 0 , 1 } x_{546} \in \B x 546 ∈ { 0 , 1 } x 551 ∈ { 0 , 1 } x_{551} \in \B x 551 ∈ { 0 , 1 } x 552 ∈ { 0 , 1 } x_{552} \in \B x 552 ∈ { 0 , 1 } x 553 ∈ { 0 , 1 } x_{553} \in \B x 553 ∈ { 0 , 1 } x 554 ∈ { 0 , 1 } x_{554} \in \B x 554 ∈ { 0 , 1 } x 555 ∈ { 0 , 1 } x_{555} \in \B x 555 ∈ { 0 , 1 } x 556 ∈ { 0 , 1 } x_{556} \in \B x 556 ∈ { 0 , 1 } x 561 ∈ { 0 , 1 } x_{561} \in \B x 561 ∈ { 0 , 1 } x 562 ∈ { 0 , 1 } x_{562} \in \B x 562 ∈ { 0 , 1 } x 563 ∈ { 0 , 1 } x_{563} \in \B x 563 ∈ { 0 , 1 } x 564 ∈ { 0 , 1 } x_{564} \in \B x 564 ∈ { 0 , 1 } x 565 ∈ { 0 , 1 } x_{565} \in \B x 565 ∈ { 0 , 1 } x 566 ∈ { 0 , 1 } x_{566} \in \B x 566 ∈ { 0 , 1 } x 611 ∈ { 0 , 1 } x_{611} \in \B x 611 ∈ { 0 , 1 } x 612 ∈ { 0 , 1 } x_{612} \in \B x 612 ∈ { 0 , 1 } x 613 ∈ { 0 , 1 } x_{613} \in \B x 613 ∈ { 0 , 1 } x 614 ∈ { 0 , 1 } x_{614} \in \B x 614 ∈ { 0 , 1 } x 615 ∈ { 0 , 1 } x_{615} \in \B x 615 ∈ { 0 , 1 } x 616 ∈ { 0 , 1 } x_{616} \in \B x 616 ∈ { 0 , 1 } x 621 ∈ { 0 , 1 } x_{621} \in \B x 621 ∈ { 0 , 1 } x 622 ∈ { 0 , 1 } x_{622} \in \B x 622 ∈ { 0 , 1 } x 623 ∈ { 0 , 1 } x_{623} \in \B x 623 ∈ { 0 , 1 } x 624 ∈ { 0 , 1 } x_{624} \in \B x 624 ∈ { 0 , 1 } x 625 ∈ { 0 , 1 } x_{625} \in \B x 625 ∈ { 0 , 1 } x 626 ∈ { 0 , 1 } x_{626} \in \B x 626 ∈ { 0 , 1 } x 631 ∈ { 0 , 1 } x_{631} \in \B x 631 ∈ { 0 , 1 } x 632 ∈ { 0 , 1 } x_{632} \in \B x 632 ∈ { 0 , 1 } x 633 ∈ { 0 , 1 } x_{633} \in \B x 633 ∈ { 0 , 1 } x 634 ∈ { 0 , 1 } x_{634} \in \B x 634 ∈ { 0 , 1 } x 635 ∈ { 0 , 1 } x_{635} \in \B x 635 ∈ { 0 , 1 } x 636 ∈ { 0 , 1 } x_{636} \in \B x 636 ∈ { 0 , 1 } x 641 ∈ { 0 , 1 } x_{641} \in \B x 641 ∈ { 0 , 1 } x 642 ∈ { 0 , 1 } x_{642} \in \B x 642 ∈ { 0 , 1 } x 643 ∈ { 0 , 1 } x_{643} \in \B x 643 ∈ { 0 , 1 } x 644 ∈ { 0 , 1 } x_{644} \in \B x 644 ∈ { 0 , 1 } x 645 ∈ { 0 , 1 } x_{645} \in \B x 645 ∈ { 0 , 1 } x 646 ∈ { 0 , 1 } x_{646} \in \B x 646 ∈ { 0 , 1 } x 651 ∈ { 0 , 1 } x_{651} \in \B x 651 ∈ { 0 , 1 } x 652 ∈ { 0 , 1 } x_{652} \in \B x 652 ∈ { 0 , 1 } x 653 ∈ { 0 , 1 } x_{653} \in \B x 653 ∈ { 0 , 1 } x 654 ∈ { 0 , 1 } x_{654} \in \B x 654 ∈ { 0 , 1 } x 655 ∈ { 0 , 1 } x_{655} \in \B x 655 ∈ { 0 , 1 } x 656 ∈ { 0 , 1 } x_{656} \in \B x 656 ∈ { 0 , 1 } x 661 ∈ { 0 , 1 } x_{661} \in \B x 661 ∈ { 0 , 1 } x 662 ∈ { 0 , 1 } x_{662} \in \B x 662 ∈ { 0 , 1 } x 663 ∈ { 0 , 1 } x_{663} \in \B x 663 ∈ { 0 , 1 } x 664 ∈ { 0 , 1 } x_{664} \in \B x 664 ∈ { 0 , 1 } x 665 ∈ { 0 , 1 } x_{665} \in \B x 665 ∈ { 0 , 1 } x 666 ∈ { 0 , 1 } x_{666} \in \B x 666 ∈ { 0 , 1 } Solving Mini Sudoku ¶ The linkedin-games library counts on MiniSudoku class, that is a child class of Sudoku
To instantiate the game, it is necessary to provide one set of inputs regarding the game in question:
filled_squaresA dictionary of square: value, where value is the digit of the filled square. With MiniSudoku class, there’s no need to inform its size or its grid block dimensions, since it’s assumed that this type of game will always be a 6 × \times × × \times × __init__ method.
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from ..base.sudoku._base import BaseSudoku
class MiniSudoku(BaseSudoku):
"""
The [LinkedIn Mini Sudoku](https://www.linkedin.com/games/mini-sudoku/) game.
A 6x6 Sudoku board with 2x3 grid blocks.
Objective:
Fill all the empty spaces in the game grid with digits from 1 to 6.
Rules:
Each row, column, and 2x3 block must be filled with a digit from 1 to 6,
without repetition in each row, column, or block.
"""
def __init__(self, filled_squares: dict[tuple[int, int], int]) -> None:
"""
Args:
filled_squares: Starting filled squares as a dictionary of `(row, column): digit` items.
"""
super().__init__(size=6, block_dims=(2,3), filled_squares=filled_squares)
Program 1: Definition of MiniSudoku class
from linkedin_games import MiniSudoku
filled_squares = {
(1,1): 1, (2,2): 2, (2,5): 3, (3,3): 3, (3,4): 6,
(4,3): 5, (4,4): 4, (5,2): 4, (5,5): 5, (6,6): 6
}
mini_sudoku = MiniSudoku(filled_squares)
The Sudoku class features the model attribute, which implements the SudokuModel object to structure the Linear Optimization logic behind the Mini Sudoku’s game.
With the model built, the public method solve() calls the restricted method _set_solution() to save the solution to _board property, which can be accessed by the public solution attribute.
With the solution obtained, the method show() plots the solved Mini Sudoku’s board.
So to solve the game and display its results, just call the public methods solve() and show(), at this order.
mini_sudoku.solve()
mini_sudoku.show()References ¶ TAKANO, Kevin; DE FREITAS, Rosiane and DE SÁ, Vinícius Gusmão. O jogo de lógica Sudoku: modelagem teórica, NP-Completude e estratégias algorítmicas exatas e heurísticas. In: CONCURSO DE TRABALHOS DE INICIAÇÃO CIENTÍFICA DA SBC (CTIC-SBC), 34., 2015, Recife. Anais […]. Porto Alegre: Sociedade Brasileira de Computação, 2015. p. 71–80.
Sudoku. Wikipedia . May 7th , 2025. Accessed on October 10th , 2025.
