The New Sudoku Players' Forum

[Maq777]

Hello,

I'm looking for more examples of orbitals with 62208 models. I have these six; does anyone know of any more?

If you have them and could share them, I would greatly appreciate it.

Best regards

Code: Select all

  Familia 62208 - A            Familia 62208 - B            Familia 62208 - C
  -------------------------    -------------------------    -------------------------
  | 1 2 3 | 4 5 6 | 7 8 9 |    | 1 2 3 | 4 5 6 | 7 8 9 |    | 1 2 3 | 4 5 6 | 7 8 9 |
  | 7 8 9 | 1 2 3 | 4 5 6 |    | 7 8 9 | 1 2 3 | 4 5 6 |    | 9 7 8 | 1 2 3 | 4 5 6 |
  | 4 5 6 | 7 9 8 | 1 2 3 |    | 4 5 6 | 7 8 9 | 1 2 3 |    | 4 5 6 | 8 9 7 | 1 2 3 |
  -------------------------    -------------------------    -------------------------
  | 3 1 2 | 6 4 5 | 9 7 8 |    | 3 1 2 | 6 4 5 | 9 7 8 |    | 3 1 2 | 6 4 5 | 8 9 7 |
  | 9 7 8 | 3 1 2 | 6 4 5 |    | 9 7 8 | 3 1 2 | 6 4 5 |    | 7 8 9 | 3 1 2 | 6 4 5 |
  | 6 4 5 | 8 7 9 | 3 1 2 |    | 6 4 5 | 8 9 7 | 3 1 2 |    | 6 4 5 | 9 7 8 | 3 1 2 |
  -------------------------    -------------------------    -------------------------
  | 2 3 1 | 5 6 4 | 8 9 7 |    | 2 3 1 | 5 6 4 | 8 9 7 |    | 2 3 1 | 5 6 4 | 9 7 8 |
  | 8 9 7 | 2 3 1 | 5 6 4 |    | 8 9 7 | 2 3 1 | 5 6 4 |    | 8 9 7 | 2 3 1 | 5 6 4 |
  | 5 6 4 | 9 8 7 | 2 3 1 |    | 5 6 4 | 9 7 8 | 2 3 1 |    | 5 6 4 | 7 8 9 | 2 3 1 |
  -------------------------    -------------------------    -------------------------
 
  Familia 62208 - D            Familia 62208 - E            Familia 62208 - F
  -------------------------    -------------------------    -------------------------
  | 1 2 3 | 4 5 6 | 7 8 9 |    | 1 2 3 | 4 5 6 | 7 8 9 |    | 1 2 3 | 4 5 6 | 7 8 9 |
  | 7 8 9 | 1 2 3 | 4 5 6 |    | 9 7 8 | 1 2 3 | 4 5 6 |    | 7 4 9 | 1 2 8 | 3 5 6 |
  | 4 5 6 | 7 8 9 | 1 2 3 |    | 6 4 5 | 9 7 8 | 1 2 3 |    | 8 5 6 | 7 3 9 | 1 2 4 |
  -------------------------    -------------------------    -------------------------
  | 8 1 2 | 6 3 5 | 9 7 4 |    | 3 1 2 | 6 4 5 | 9 7 8 |    | 3 1 7 | 2 4 5 | 9 6 8 |
  | 9 7 4 | 8 1 2 | 6 3 5 |    | 8 9 7 | 3 1 2 | 6 4 5 |    | 9 6 4 | 8 1 7 | 2 3 5 |
  | 6 3 5 | 9 7 4 | 8 1 2 |    | 5 6 4 | 8 9 7 | 3 1 2 |    | 2 8 5 | 9 6 3 | 4 1 7 |
  -------------------------    -------------------------    -------------------------
  | 2 4 1 | 5 6 8 | 3 9 7 |    | 2 3 1 | 5 6 4 | 8 9 7 |    | 6 3 1 | 5 7 4 | 8 9 2 |
  | 3 9 7 | 2 4 1 | 5 6 8 |    | 7 8 9 | 2 3 1 | 5 6 4 |    | 4 9 2 | 6 8 1 | 5 7 3 |
  | 5 6 8 | 3 9 7 | 2 4 1 |    | 4 5 6 | 7 8 9 | 2 3 1 |    | 5 7 8 | 3 9 2 | 6 4 1 |
  -------------------------    -------------------------    -------------------------


[Serg]

Hi, Maq777!
What does it mean "62208 orbital model"? What property of Sudoku solution grids do you mean?

Serg

[Maq777]

Hi, Maq777!
What does it mean "62208 orbital model"? What property of Sudoku solution grids do you mean?

Serg

Hi Serg,

I'm referring to this line which indicates that there are 11 different orbits with 62208 patterns. I have already identified 6 of those 11 orbits, and I was wondering if anyone has already identified Sudoku examples of the other 5 that I am missing.

Code: Select all

#ORBIT   COUNT       MODELS
1        1           5184
2        1           20736
3        3           31104
4        2           46656
5        11          62208  <----
6        15          93312
7        2           124416
8        85          186624
9        92          279936
10       42          373248
11       29          419904
12       1257        559872
13       2826        839808
14       7336        1119744
15       548449      1679616
16       5472170387  3359232


[champagne]

like 'Serg' still don't see what is the property

[Serg]

Hi, Maq777!
If I understood you correctly, you want to get the list of Sudoku solution grids having exactly 54 (3359232/62208) automorphisms under the group of VPT (validity preserving transformations, such as swapping rows in a band, swapping stacks, transposing, etc.). You expect to find 11 such solution grids, but found 6 of them only, right?

Serg

[Maq777]

Hi, Maq777!
If I understood you correctly, you want to get the list of Sudoku solution grids having exactly 54 (3359232/62208) automorphisms under the group of VPT (validity preserving transformations, such as swapping rows in a band, swapping stacks, transposing, etc.). You expect to find 11 such solution grids, but found 6 of them only, right?

Serg

Exactly !!!


I've just found one more.

Code: Select all

62208 - G
-------------------------
| 1 2 3 | 4 5 6 | 7 8 9 |
| 7 8 9 | 1 2 3 | 4 5 6 |
| 4 5 6 | 7 8 9 | 1 2 3 |
-------------------------
| 6 1 8 | 9 4 2 | 3 7 5 |
| 3 7 5 | 6 1 8 | 9 4 2 |
| 9 4 2 | 3 7 5 | 6 1 8 |
-------------------------
| 5 9 1 | 2 6 7 | 8 3 4 |
| 8 3 4 | 5 9 1 | 2 6 7 |
| 2 6 7 | 8 3 4 | 5 9 1 |
-------------------------


[Serg]

Hi, Maq777!
Here is the list of automorphic Sudoku solution grids having 27 or more automorphisms (posted by gsf in 2007).

Serg

[Maq777]

Serg,

Thank you so much, this was exactly what I was looking for.

[Maq777]

Hi all! Please follow the link below to explore the interactive 3D visualizer:

https://www.youtube.com/watch?v=-rLcmRIOPXk&list=PLI40n2bVm44Re6y-qxrE4BLiRXRTvz_vy&index=25

Welcome to the geometric core of the Sudoku manifold. This interactive 3D physics simulator projects the complex ordinal data of the "62208 Family" into a navigable space, translating raw combinatorial data into pure structural architecture.

In this force-directed graph, we are looking at the foundational building blocks of the hypercube. Every spherical node represents a unique Template (a strictly defined grid), identified by its precise Quinteiro Ordinal. When nine specific templates perfectly harmonize to create a valid mathematical "Face" within the hyperspace, they form a Model. These profound geometric bonds are represented by the tension edges, pulling the nodes together into tight, unbreakable cliques.

The Analytical Imperative: It is strictly indispensable to work with and analyze these individual templates to truly understand the internal structure of Sudoku. Treating the game merely as a puzzle or a brute-force computing challenge blinds us to its true nature. The templates are the fundamental geometric vertices of a 6D polytope. By dissecting models down to their constituent templates, we can observe the "bridge nodes" that link entirely different mathematical families together. We stop looking at isolated lists of numbers and begin to map the rigid, crystalline skeleton of the hypercube.

At the absolute center of this visualization lies Grid 1 (The Universal Pivot or "Kilometer Zero"). Because this specific template is present across all models within this analyzed cluster, its gravitational pull anchors the entire ecosystem, forming a perfectly balanced, multi-dimensional geodesic sphere.

By utilizing the Quinteiro gravity controls, interactive family filters, and immersive Zen mode, researchers can isolate these orbital clusters in real-time. This visualizer is not just a data display; it is the definitive proof that the chaotic permutations of Sudoku are actually the shadows of an exquisitely ordered, universal architecture. It is our primary compass in the quest for the ultimate mathematical grail: the Universal Formula $f(n)$.

[coloin]

Thats a very nice demonstration of "sudoku space" even if it is just for 11 sudoku solution grids ....

By dissecting models down to their constituent templates

Perhaps you could run through the templates of Grid 1 and outline the mechanism of the 54 automorphic transformations

How are there 5 orbits in these 11 Essentially different solution grids ?

Code: Select all

123456789456789123789123456231564897564897231897231564348672915672915348915348672 # 54 001 964568
123456789456789123789123456231564897564897231897231564348915672672348915915672348 # 54 001 964569
123456789456789123789123456234567891567891234891234567318642975642975318975318642 # 54 001 992761
123456789456789123789123456234567891567891234891234567345678912678912345912345678 # 54 001 992769
123456789456789123789123456234567891567891234891234567372615948615948372948372615 # 54 001 992773
123456789456789123789123456235964817817235964964817235392641578578392641641578392 # 54 001 1006748
123456789456789123789123456267591348591834672834267915375618294618942537942375861 # 54 001 1007164
123456789456789123789123456267591834591834267834267591375942618618375942942618375 # 54 001 1007169
123456789456789123897231564231564978564978231789312645312645897645897312978123456 # 54 016 473714886
123456789456789231789312456231645978645978312978123645312564897564897123897231564 # 54 030 1062073983
123456789457289163689173452235964817816735924974812635392641578568397241741528396 # 54 348 5472677656

morphed to best match for thr first grid

Hidden Text: Show

Code: Select all

similarity 63                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
123456789456789123789123456231564897564897231897231564348915672672348915915672348
similarity 63                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
123456789456789123789123456234567891567891234891234567318642975642975318975318642
similarity 63                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
123456789456789123789123456234567891567891234891234567345678912678912345912345678
similarity 63                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
723156489156489723489723156231564897564897231897231564342675918675918342918342675
similarity 51                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
123456789456789123789123456235964817964817235817235964392641578578392641641578392
similarity 51                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
429753861753186294186429537231564789564897123897231456348672915672915348915348672
similarity 63                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
123456789456789123789123456294861537861537294537294861348672915672915348915348672
similarity 57                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
423756189186429753759183426291534867564897231837261594948372615672915348315648972
similarity 57                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
723156489456789123189423756231564897567891234894237561342675918978312645615948372
similarity 39                                                                   
123456789456789123789123456231564897564897231897231564348672915672915348915348672
123476589456389127789125436971562843362847951845931762238714695697258314514693278


How are there 62208 models ? ....

[Maq777]

Hi Coloin,

Calculating the templates that make up your first load line, I get:
123456789456789123789123456231564897564897231897231564348672915672915348915348672 # 54 001 964568

Template1.Template2.Template3.Template4.Template5.Template6.Template7.Template8.Template9
2688.8955.15356.15716.21990.28389.33981.40250.46656

Now, I run the 3,359,232 transformations on this model and keep only the distinct ones that contain Template 1, yielding:

Complete_Discovered_Family
1.6403.12673.18562.24952.30364.33029.36827.43121
1.6403.12673.18844.24376.30658.32177.38555.42245
1.6403.12673.19144.25522.29212.32471.38837.41669
1.6403.12673.19438.25804.28636.31595.37985.43397
1.6403.12673.19996.23794.30088.33605.36533.42839
1.6403.12673.20572.23500.29806.31877.37409.43691
1.6406.12677.18268.24671.30937.31301.37699.43972
1.6407.12676.18268.24667.30941.31301.37702.43969
1.6523.12865.18268.24742.30820.31301.37511.43901
1.6526.12869.18268.24743.30817.31301.37507.43900
1.6595.12793.18268.24550.31012.31301.37631.43781
1.6599.12796.18268.24547.31013.31301.37630.43777

Since I get 12 different ones and I know that each corresponds to a group of 5184, I get:

5184 * 12 = 62208
For me, 12 (12 lines of models) is how I find 62208 because what I do is multiply by 5184 models.

3359232 / 54 = 62208
For you, 54 is how you find 62208 because you divide the geometric transformations by it.

Same result, different approaches.

Each COLOR in the graph represents a group of these twelve models. They all have Template 1 in common, and they all form orbits of solutions with 62,208 models, but each color is disjoint from the others.

What you are seeing are all the models of the solutions with orbits of 62,208 models that share Template 1.

And from Jarvis's work, we know there are 11 of these groups, all of which Serg helped me find.

So, each color is one of those 11 groups, which is why it goes from Fam1 to Fam11.

Since you are giving me 11 load lines, if I apply the same procedure to each one, I get 12 distinct models in each case.

The templates I show in the graph are all of those, and I connect them with edges if they can be combined to form a solution model.

Does that make sense?

[coloin]

almost...
I see that 6^8 x 2 = 3359232 for the non-automorphic grids at #16 and the MC grid [#1] has the 5184 number ...
I see that you have labelled the 46646 templates.
How do you know that each corresponds to a group of 5184 ?

[Maq777]

Hello Coloin,

The progression of solution groups in increments of 5,184 is not contingent; it is structural. In 9x9 Sudoku, the number 5,184 is the symmetry seed (the order of the board's automorphism subgroup that preserves the block structure).

1. Family Invariance

A model cannot exist in isolation. By applying a geometric transformation to a Template, the only thing you can obtain is another template.

In fact, if you apply all transformations to a single template, you will find that it transforms 72 times into each of the other 46,656.

If you correctly choose the nine templates of the model, then you can create the family of 5,184 models:

(Stabilizer 648 ----> 3,359,232 / 648 = 5,184)
(For me 1 * 5,184 = 5,184)

-- 5,184
1.6403.12673.18268.24670.30940.31301.37703.43973

72 + 72 + 72 + 72 + 72 + 72 + 72 + 72 + 72 = 72 * 9 = 648 (Stabilizer of the smallest family "Seed"). This is the ORIGIN of the stabilizer.

When applying the group of 3,359,232 geometric transformations, the model necessarily moves toward other states within its own equivalence class. The closure of the system dictates that a model can only transform into another member of the same family.

2. The Minimum Unit (5,184)

When nine templates are selected with mathematical harmony (orthogonality in the hypercube), the resulting configuration generates a primary family of 5,184 models.

In this unit: Each template is used exactly once. There is a unique occurrence of Template 1 (Kilometer Zero). Any operation within this subgroup maps the set onto itself.

3. Derivation of the 62,208 Model

The number 62,208 is simply the aggregation of these fundamental units.

When analyzing the orbits, we observe that: 3,359,232 / 54 = 62,208. From the seed perspective: 12 * 5,184 = 62,208.

This demonstrates that what we call "Model 62,208" is actually a hyper-cluster composed of 12 base families of 5,184 models each, linked by the co-presence of Template 1.

It is not a random accumulation; it is the exact partition of the orbital space under the Quinteiro metric.

Finally, I show you other examples:

(Stabilizer 162 ----> 3,359,232 / 162 = 20,736)
(For me 4 * 5,184 = 20,736)
-- 20,736
1.6598.12797.18268.24551.31009.31301.37627.43780
1.6403.12673.19720.25228.28930.33323.37109.42545
1.6403.12673.20290.24076.29512.32753.38261.41963
1.6527.12868.18268.24739.30821.31301.37510.43897

(Stabilizer 108 ----> 3,359,232 / 108 = 31,104)
(For me 6 * 5,184 = 31,104)
-- FAMILY 31,104 - A
1.6403.12673.18268.24670.30940.31367.37637.44039
1.6403.12673.18334.24604.31006.31301.37703.43973
1.6403.12673.18334.24604.31006.31367.37637.44039
1.6403.15290.18268.24670.28323.31301.37703.46590
1.9020.12673.18268.22053.30940.31301.40320.43973
1.9020.15290.18268.22053.28323.31301.40320.46590
-- FAMILY 31,104 - B
1.6403.13521.17040.24670.30940.31301.40766.43973
1.6403.13986.18268.21584.30940.35757.37703.43973
1.8508.12673.17804.24670.30940.31301.37703.45015
1.8508.13986.17804.21584.30940.35757.37703.45015
1.9783.12673.18268.24670.26292.35252.37703.43973
1.9783.13521.17040.24670.26292.35252.40766.43973
-- FAMILY 31,104 - C
1.6403.12673.18269.24671.30941.31300.37702.43972
1.6403.12673.18340.24742.31012.31229.37631.43901
1.6403.12673.18341.24743.31013.31228.37630.43900
1.6685.12391.18268.25522.30088.31301.38837.42839
1.7255.11821.18268.25804.29806.31301.37985.43691
1.7537.11539.18268.24952.30658.31301.38555.43121

I am currently working to publish a more detailed and formal explanation on arXiv.org.

Title:
"A Structural Characterization of Sudoku via Templates and Compatibility Graphs"

I am still looking for someone to provide the initial endorsement; I have been studying Sudoku very deeply for 15 years now.

Check out the following link as well:
https://www.youtube.com/watch?v=SB2jTqi5I5U&list=PLI40n2bVm44Re6y-qxrE4BLiRXRTvz_vy&index=22

And while you're at it, feel free to take a look at my entire list of Sudoku videos.

Best regards.

[coloin]

Certainly those visualizations are what I imagined the "sudoku puzzle space" to maybe start to look like !
This brings to reality some of the thoughts in this thread Sudoku Space

random puzzles /grids tend to share at least a few clues.. but as you suggest in your presentation this would be a supernova on the screen

Code: Select all

Average similarity for n clues 
18 clues
similarity 10
...5..87...9.......2.......7....9.....4.1.6.........8..7.......5.....2....13..4.9
...5.48...89.........6.....7......5.3...1.6.........8.........1....9.2..5..3....9

30 clues
similarity 14
2.......9..7....1....2.8.4.8.3..2.719......361..6..9.8.1.72.....2.8.9.6.7965.....
.1....2.9.64.9.3.........4..........9...7..361.6.429.8..172.8..62..59.7.79..8....

81 clues
similarity 49
136592874859147326427836915712689543384715692695423781973254168548961237261378459
326591784519487326487632915792168543138745692654329871973254168845916237261873459

however Im still stuck on the actual transformations which add up to 5184 which of the 6^8 x 2 are these ?

but i think i am getting there
this puzzle has a 123456789 transversal [template] at the end of the sudoku space post [ above]

Code: Select all

+---+---+---+
|1..|..4|...|
|...|29.|..5|
|..6|...|3..|
+---+---+---+
|.4.|...|...|
|2..|.5.|1..|
|8..|...|.6.|
+---+---+---+
|..7|...|..4|
|...|..8|...|
|.5.|...|..9|
+---+---+---+

Indeed it has 72 isomorphs which maintain the transversal [9x2x2x2=72]

[Maq777]

Hi Coloin,

Great question — the number 5184 turns out to be a fundamental structural quantity when you look at Sudoku through single-digit placements (what I call templates).

https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/0a.-%20TEMPLATE%20-%201.png

In a standard 9×9 Sudoku, a template is a valid placement of a single digit: exactly one occurrence in each row, column, and region. The total number of such templates is:

|G| = (3!)^6 = 46656.

Each template occupies exactly 9 cells, so the total number of template–cell incidences is:

46656 × 9 = 419904.

Since the Sudoku grid is completely symmetric (no cell is structurally distinguished), each of the 81 cells must belong to the same number of templates. Therefore:

419904 / 81 = 5184.

So 5184 is the number of templates that pass through any given cell. I think of this as a kind of “structural unit” of the system.

Before adding a template
https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/1a.-%20AntesDeColocarUnTemplate%20I.png

After adding a template
https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/1a.-%20DespuesDeColocarUnTemplate%20II.png
---

Where this becomes more interesting is when you look at Sudoku solutions up to symmetry.

If you consider the group of valid Sudoku transformations (row/column permutations respecting structure, plus transposition), its size is:

|G_geo| = 3,359,232.

This group acts on the set of Sudoku solutions, partitioning them into orbits. Each orbit corresponds to a family of structurally equivalent solutions.

By the orbit–stabilizer relation:

|O(x)| = |G_geo| / |Stab(x)|.

What I observed empirically is that orbit sizes are not arbitrary — they appear as multiples of 5184:

5184, 20736, 31104, ..., 3,359,232.

This suggests that 5184 plays a deeper role than just a counting artifact: it behaves like a base scale for the decomposition of the solution space.

---

From a structural point of view, there are three interacting layers:

1. Templates (single-digit placements),
2. Compatibility (which templates can coexist),
3. Group action (which organizes solutions into orbits).

A Sudoku solution can be seen as a set of 9 mutually compatible templates (one per digit), i.e., a clique in a compatibility graph.

Before adding a template
https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/1b.-%20AntesDeColocarUnTemplate%20III.png

After adding a template
https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/1b.-%20DespuesDeColocarUnTemplate%20IV.png

What is particularly interesting is that this global organization seems to be already encoded locally: if you build solutions incrementally by adding compatible templates, the structure “self-organizes,” and the orbit decomposition emerges naturally without explicitly applying group transformations.

https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/1ba.-%20CompatibilidadTemplates.png

---

So in short:

- 5184 comes from uniform distribution of templates over cells,
- orbit sizes are multiples of 5184,
- and this reflects a deeper interaction between symmetry and compatibility.

Before adding a template
https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/1c.-%20AntesDePonerPrimerTemplate.png

After adding a template
https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/1c.-%20DepuesDePonerPrimerTemplate.png

Exactly the same thing happens in 4x4 Sudoku, but this time the seed is 4 instead of 5184...
- 4 templates pass through each square
- the orbits of the templates are growths with multiples of 4

I’m still working on formalizing exactly how far this goes, but the pattern is very consistent across the data.

Best,
Mike