The New Sudoku Players' Forum

[champagne]

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

I did not follow all the topic, but clearly something to do with auto morphs in a solution grid.

3 359 232 / 5184 = 648

one solution grid in the min lexical catalog and only one has this number of iso morphs

[Maq777]

I've previously previewed some of this in these forum threads:

http://forum.enjoysudoku.com/shidoku-study-with-graphs-t38842.html

Just as 4x4 sudoku can be perfectly represented in a hypercube with 16 vertices for its templates, 9x9 sudoku can be represented in its hypercube equivalent of 46656 vertices for its templates, on wikipedia you have it as Y6 / 6.

https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/Hipercuboy66.png

http://forum.enjoysudoku.com/shidoku-study-with-graphs-t38842.html

http://forum.enjoysudoku.com/sudoku-is-a-6-dimensional-problem-t2151.html

I think the dimensionality of the sudokus would be like this

Code: Select all

Sudoku   Templates       n       P       R(dimension)    Vertices
-------------------------------------------------------------------------
1x1      1               1       1       1               1
4x4      16              2       2       4               16
9x9      46656           3       6       6               46656
16x16    110075314176    4       24      8               110075314176
25x25    6,19174E+20     5       120     10              6,19174E+20
36x36    1,94084E+34     6       720     12              1,94084E+34
-------------------------------------------------------------------------


.- For n = 2 we are in the case of sudoku 4x4, 16 vertices are needed for its 16 Templates, the way to build it is:
2 * 2 * 2 * 2 = 2 ^ 4 = 16 that is why the 4x4 sudoku puzzle is located at R4 and at P = 2
With this we can embed the 4x4 sudoku in the first tesseract (the usual Y (2-4)), for that we already know that it has 24 faces of which are used
only 12 to place the possible models or invariant grids of that sudoku puzzle

.- For n = 3 we are in the case of sudoku 9x9 (the typical one of the newspapers) 46656 vertices are needed for its 46656 Templates, the way to build it is:
6 * 6 * 6 * 6 * 6 * 6 = 6 ^ 6 = 46656 that is why the 9x9 sudoku puzzle is located at R6 and at P = 6
With this we can embed the sudoku 9x9 in the generalized hypercube Y (6-6), for that we do not know how many 9-faces it has,
but we know that there must be at least 18,383,222,420,692,992 which are used to place the possible models or invariant grids of the 9x9 sudoku

.- For n = 4 we are in the case of sudoku 16x16, 110075314176 vertices are needed for its 110075314176 Templates, the way to build it is:
24 * 24 * 24 * 24 * 24 * 24 * 24 * 24 = 24 ^ 8 = 110075314176 that is why the 16x16 sudoku is located at R8 and at P = 24
With this we can embed the 16x16 sudoku in the generalized hypercube Y (24-8), for that we do not know how many 16-faces it has,
nor do we know the order of magnitude of the possible 16x16 sudoku models

We can continue like this successively Embedding the larger and larger sudokus into bigger and bigger Politopes .

Visualizing the full space of Sudoku 9x9 models and solutions requires rendering 46656 nodes at once. With each node having a degree of 17972, we end up with over 419 million edges.

That’s simply too dense to visualize effectively, so I decided to focus on partial views. I chose to center these on 'Template 1'—an arbitrary choice, as any other template would have yielded the same results.

However, I did manage to visualize all models and solutions at once using a Tesseract for the 4x4 Sudoku. Since it only has 16 nodes, I mapped a template to each node and linked those that were compatible with edges.

With the tesseract structure, I can run the 128 geometric transformations to see how the system moves and shifts, ensuring the orbits of the models stay within their solution families.

Mike

[Maq777]

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

I’m glad I could contribute! We’re all here to help each other unravel the mysteries of Sudoku...

[Maq777]

Coloin,

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]

PROBLEM :

Code: Select all

1....4......29...5..6...3...4.......2...5.1..8......6...7.....4.....8....5......9

SOLUTION :

Code: Select all

182534976734296815596871342941683527263457198875129463317965284629348751458712639

PROVIDED MODEL: 4106.7914.11189.20098.21763.27990.34881.36854.46485

CANONICAL MODEL: 1.6412.12873.18954.24464.30686.33527.37197.41946

CANONICAL SOLUTION:

Code: Select all

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

Observations:
This input lines belong to orbits containing 3,359,232 models.
This is because, I obtain one occurrence of Template 1 out of 648.
648 × 5,184 = 3,359,232 models

Where: the provided model was the model you gave us, and the canonical model was the model I calculated.

Both models are part of the same orbit, which consists of 3,359,232 models, of which these two are examples.

A canonical model is the model within the orbit that uses the minimum templates (the smallest ones in my numbering system).

Mike

[Maq777]

Hi all,

We have also prepared a paper for when we are ready to publish on arXiv.org. Here is a brief look at the beginning of the document:

Towards an Algebraic–Geometric Characterization of Sudoku Templates, Group Actions, and Emergent Structure in Discrete Spaces

Miguel Angel Quinteiro Fernandez --- Independent Researcher
Miguel Angel Quinteiro Pinero --- Independent Researcher

Abstract

Sudoku is typically studied as a problem of enumerating solutions. However, this perspective obscures its underlying combinatorial structure.

In this work, we propose a reformulation based on templates (single-digit configurations), their compatibility graph, and the action of transformation groups. We show that:

• the system is structurally uniform,
• solutions correspond to maximal cliques,
• models organize into orbits under group action.

Additionally, we introduce the principle that global Sudoku structure emerges from local compatibility constraints, suggesting that orbit decomposition is an intrinsic property of the system.

1 Introduction

Sudoku of order n^2 is a highly constrained combinatorial system. Although the number of solutions is known in some cases, no complete structural theory explains their origin.

This work proposes a framework based on:
• hierarchical decomposition,
• global symmetry,
• emergent dynamics.

2 Templates

Definition 1. A template is a subset of cells such that:
• exactly one cell is selected in each row,
• exactly one cell is selected in each column,
• exactly one cell is selected in each region.

Proposition 1. The number of templates is:

|G| = (n!)^2n

I'll leave it at that for now. I hope this provides more context on our work, and I'm happy to discuss any questions you might have!

Best regard