A General Logic for Sudoku

Golden Nugget



Points. A logical solution to Golden Nugget gives insight into the nature of extreme puzzles. The solution uses a freeform logic that can assume any form. Many of GN's eliminations tkae the form of layered single digit patterns that connect vertically through common groups of linksets.


Golden Nugget




The Solution


Golden Nugget is without a doubt a difficult puzzle. This set logic solution provides perhaps a different way of viewing the logic inside of the puzzle as compared to other methods. Below is a chart that shows the relative difficulty of each move based on (blue) the number of sets used by the logic and (red) the number of nodes involved in the logic. Most difficult moves require around 20 - 25 sets and roughly 70 nodes. A few eliminations have some broken wing logic, which uses a relatively higher number of sets. The most difficult eliminations fall into 3 groups of 10 to 15 eliminations where each group is followed by an assignment. These assignments roughly correlate to the gaps between the 3 groups.

Like the solution to Easter Monster, the solution to Golden Nugget uses a freeform logic that can take on most any form. The first thing apparent from the solution is that many difficult eliminations are well beyond any simple interpretation using set logic rules.


However, a number of interesting exceptions take the form of 4 or 5 single digit layers connected through a single group of vertical linksets. Each layer forms a strong inference set (SIS) that combines with other SIS in the vertical linksets to cause eliminations. In some cases, one layer receives the combined effect of the other layers and in one case, the SIS fill the vertical linksets until they become rank 0 and cause eliminations. Such highly organized logic is suprising and may indicate a logical weakness in this monster. (Note: the solver works based on sets and permutations without using rules like rank, cover sets, or alternating strong-weak links, any organized structure is what the solver considered easy to find.)


The table at the bottom of the page gives links to  set logic and diagrams for all eliminations. The thumbs in two tables below link to specific eliminations that have been studied in more detail, which will be added to as more are analyzed.



Early Eliminations


Layer Cake

E1. Five single digit layers combine 5 strong inference sets to make rank 0 logic.


The Worst

E16. The worst elimination required 65 sets + linksets and eliminates two candidates.


Negative Inference Set

E19. Three layers form 3 strong inference sets that combine into a 4th layer.




The Tail End


Simple Proof

E10. An example of a simple set logic proof for moderately complex logic.


Simple Proof

E64. A second example of a simple proof with set logic.

One Rank 0 Linkset

E39 uses one elimination to build the logic for a second elimination.  No analysis.


In the shadow of triplets

E44. Rank 0 logic with several triplets that can raise rank, but not at the elimination.

Rank 4 + 3 Triplets

E23x. Rank 4 example with 3 triplets to lower rank to 1 for an elimination.


E49. Simple Rank 2 ALS and chain example with triplet.


E51x True to life, this simple 18 sided bi-value nice loop shows

Nowhere Loop

E64. True to life, this simple 18 sided bi-value nice loop shows how hard it is to get anywhere.

Final Breakup

E67. The final breakup of GN's logic.





List of Links to Eliminations 1 - 67


G01a, grid[21.242] 58 sets s2:0 p136 p793

G33a, grid[23.196] 53 sets s1:0 p571

G01b, grid[21.242] 38 sets s1:0 p793

G34a, grid[23.195] 46 sets s1:0 p672

G02a, grid[21.240] 47 sets s1:0 p743

G35a, grid[23.194] 43 sets s2:0 p277 p367

G03a, grid[21.239] 58 sets s1:0 p246

G36a, grid[23.192] 11 sets s1:0 p791

G04a, grid[21.238] 47 sets s1:0 p217

G37a, grid[23.191] 11 sets s2:0 p392 p982

G05a, grid[21.237] 54 sets s1:0 p443

G38a, grid[23.189] 3 sets s1:0 p472

G06a, grid[21.236] 61 sets s1:0 p284

G39a, grid[23.188] 13 sets s1:0 p252

G07a, grid[21.235] 51 sets s1:0 p242

G39b, grid[24.128] 12 sets s1:0 p252

G08a, grid[21.234] 50 sets s1:0 p366

G40a, grid[23.188] 31 sets s1:0 p463

G09a, grid[21.232] 46 sets s1:0 p797

G41a, grid[23.186] 13 sets s2:1

G09b, grid[21.232] 56 sets s1:0 p282

G42a, grid[24.180] 4 sets s1:0 p156

G10a, grid[21.230] 20 sets s1:0 p236

G43a, grid[24.179] 13 sets s1:0 p953 ]

G11a, grid[21.229]30 sets s1:0 p226

G44a, grid[24.177] 14 sets s1:0 p546

G12a, grid[21.228] 53 sets s2:1 p287 p386 a286

G45a, grid[24.176] 24 sets s1:0 p774

G13a, grid[21.227] 2 sets s2:1 p216 p256

 G46a, grid[24.175] 8 sets s1:0 p864

G14a, grid[22.223] 43 sets s1:0 p177

G47a, grid[24.174] 9 sets s1:0 p594

G15a, grid[22.222] 55 sets s1:0 p972

G48a, grid[24.173] 9 sets s1:0 p766

G16a, grid[22.221] 65 sets s1:0 p126

G49a, grid[24.172] 16 sets s1:0 p474

G17a, grid[22.220] 61 sets s1:0 p692

G50a, grid[24.171] 13 sets s1:0 p124

G18a, grid[22.219] 44 sets s1:0 p772

G51a, grid[24.170] 31 sets s2:1 p677 p717 a777 

G19a, grid[22.218] 32 sets s1:0 p142

G52a, grid[25.163] 7 sets s1:0 p425

G20a, grid[22.217] 48 sets s1:0 p319

G53a, grid[26.154] 8 sets s4:2 p171 p741 p831 p931

G21a, grid[22.216] 60 sets s1:0 p212

G54a, grid[28.143] 4 sets s2:0 p384 p394

G22a, grid[22.215] 44 sets s1:0 p248

G55a, grid[28.142] 10 sets s3:2 p112 p817 a117

G23a, grid[22.214] 2 sets s2:0 p118 p128

G56a[31.133] 2 sets s2:0 p384 p884

G24a, grid[22.212] 40 sets s3:1 p224 p249 p253 a243

G57a[31.131] 2 sets s1:0 p394

G25a, grid[23.206] 2 sets s1:0 p329

G58a[31.130] 9 sets s1:0 p663

G26a, grid[23.205] 60 sets s2:0 p116 p346

G59a[31.129] 2 sets s1:0 p853

G27a, grid[23.203] 41 sets s1:0 p516

G60a[31.128] 13 sets s1:0 p991

G28a, grid[23.202] 45 sets s1:0 p134

G61a[31.127] 7 sets s1:0 p963

G29a, grid[23.201] 43 sets s2:0 p921 p925

G62a[31.126] 13 sets s3:0 p665 p951 p966

G30a, grid[23.199] 37 sets s1:0 p535

G63a[31.123] 2 sets s2:0 p881 p891

G31a, grid[23.198] 25 sets s1:0 p164

G64a[31.121] 18 sets s1:0 p553

G32a, grid[23.197] 43 sets s1:0 p382

G65a[31.121] 2 sets s1:0 p845

G66a[31.120]16 sets s3:3 p397 p591 p698 a387 a391 a697

G67a [End] 25 sets s16:10 p146 p312 p326 p349 p424 p548 p588 p593 p623 p688 p716 p869





For more information about notation and how set logic works see Quick Guide to Sudoku Set Logic. To look for specific Sudoku examples, go back to the Homepage.