Shidoku Study with Graphs

Everything about Sudoku that doesn't fit in one of the other sections

Shidoku Study with Graphs

Shidoku Study with Graphs,

Here I present part of the study of Shidoku using Graphs

The representation of templates and models in the tesseract remains this way

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

Greetings

Graphs 01
Attachments
Sudoku4x4(Reducido02).png (138.12 KiB) Viewed 463 times
Sudoku4x4(Reducido01).png (180.19 KiB) Viewed 463 times
Last edited by Maq777 on Thu Mar 25, 2021 2:44 pm, edited 1 time in total.
Maq777

Posts: 52
Joined: 30 April 2016

Re: Shidoku Study with Graphs

Greetings

We have 16 cells in total, with the meeting of 4 cells we make a Template, and we can make only 16 diferent templates

With the meeting of 4 templates we make a model or solution grid

In total we can make only 12 models, 4 red faces of the tesseract form family 1

And 4 Yellow faces joined to 4 Blue faces generate family 2.

The whole tesseract works together and it is where we can see at the same time all the geometric transformations and all the relabels on all the 4x4 sudoku solutions.

best regards
Miguel Quinteiro

Some Images.
Attachments
Sudoku 4x4
SudokuCompleto.png (100.85 KiB) Viewed 432 times
Family 02
Familia02(reducida).png (17.76 KiB) Viewed 432 times
Family 01
Familia01(reducida).png (11.26 KiB) Viewed 432 times
Maq777

Posts: 52
Joined: 30 April 2016

Re: Shidoku Study with Graphs

What has been said can be better appreciated in the following video

Maq777

Posts: 52
Joined: 30 April 2016

Re: Shidoku Study with Graphs

To give an explanation of the images shown

We have 16 cells on the 4x4 sudoku board, numbered from 1 to 16 from top to bottom and from left to right

taking four cells that do not collide neither in row, nor in column, nor in region we form a template, to place the same digit inside, in this way we generate the following 16 templates

----------------- Cells Number
Template 01 : {01, 07, 10, 16}
Template 02 : {01, 07, 12, 14}
Template 03 : {01, 08, 10, 15}
Template 04 : {01, 08, 11, 14}
Template 05 : {02, 07, 09, 16}
Template 06 : {02, 07, 12, 13}
Template 07 : {02, 08, 09, 15}
Template 08 : {02, 08, 11, 13}
Template 09 : {03, 05, 10, 16}
Template 10 : {03, 05, 12, 14}
Template 11 : {03, 06, 09, 16}
Template 12 : {03, 06, 12, 13}
Template 13 : {04, 05, 10, 15}
Template 14 : {04, 05, 11, 14}
Template 15 : {04, 06, 09, 15}
Template 16 : {04, 06, 11, 13}

Now taking four of those templates and putting them together until filling the board we form the following 12 models

------------ Templates Number
Model 01 : {01, 07, 10, 16}
Model 02 : {01, 07, 12, 14}
Model 03 : {01, 08, 10, 15}
Model 04 : {02, 07, 09, 16}
Model 05 : {02, 08, 09, 15}
Model 06 : {02, 08, 11, 13}
Model 07 : {03, 05, 10, 16}
Model 08 : {03, 05, 12, 14}
Model 09 : {03, 06, 11, 14}
Model 10 : {04, 05, 12, 13}
Model 11 : {04, 06, 09, 15}
Model 12 : {04, 06, 11, 13}

In this way we generate the following graph
https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/GrafoCellsTemplatesModels.png
Maq777

Posts: 52
Joined: 30 April 2016

Re: Shidoku Study with Graphs

Now removing the cells from the graph and changing the position of the templates to form the tesseract, the models are associated with the faces of the tesseract.

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

Being the Vertical faces of the hypercube (red color) the models 01, 05, 08, 12

https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/Mod01050812.png

The horizontal faces (Yellow) models 02, 06, 11, 07

https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/Mod02060711.png

And the oblique faces (blue faces) models 03, 04, 09, 10

https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/Mod03040910.png

By applying the 128 geometric transformations to the models, we can switch between the models 01, 05, 08, 12 by exchanging the red faces with each other.

At the same time applying the 128 transformations we can exchange the models 02, 03, 04, 06, 07, 09, 10, 11 with each other
Maq777

Posts: 52
Joined: 30 April 2016

Re: Shidoku Study with Graphs

And as I say here

Maq777 wrote: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

We could do the same for 9x9 sudoku in a higher member of the hypercube family. A hypercube that has 46,656 vertices where we can place each of its templates, Of course here as Jarvis and Russell taught us we would have to accommodate 18.383.222.420.692.992 models on the faces formed by nine 9 of the vertices.
Last edited by Maq777 on Fri Mar 26, 2021 6:24 pm, edited 1 time in total.
Maq777

Posts: 52
Joined: 30 April 2016

Re: Shidoku Study with Graphs

Maq777 wrote:Greetings

We have 16 cells in total, with the meeting of 4 cells we make a Template, and we can make only 16 diferent templates

With the meeting of 4 templates we make a model or solution grid

In total we can make only 12 models, 4 red faces of the tesseract form family 1

And 4 Yellow faces joined to 4 Blue faces generate family 2.

The whole tesseract works together and it is where we can see at the same time all the geometric transformations and all the relabels on all the 4x4 sudoku solutions.

best regards
Miguel Quinteiro

I really find the content very much helpful to me. Thanks for sharing with us.

Some Images.
niamul21

Posts: 1
Joined: 26 March 2021

Re: Shidoku Study with Graphs

Some characteristics to look for in the hypercube of 46,656 vertices are:

Each vertex must be connected by edges to other 17,972 neighboring vertices.

Taking neighboring vertices nine by nine, 18,383,222,420,692,992 faces should be formed.

5184 of these faces must be parallel and applying the 3,359,232 geometric permutations to move the vertices around the hypercube, the 5184 faces must be able to interchange with each other.

For the rest of the faces, the pattern already described previously in this forum must be followed.

Code: Select all
`++++++++++++++ Parallel faces always in multiples of 5184=========================================================   1 5,472,170,387      18,382,289,873,462,784   2       548,449      921,183,715,584    3         7,336      8,214,441,984    4         2,826      2,373,297,408    6         1,257      703,759,104    8            29      12,177,216    9            42      15,676,416   12            92      25,754,112   18            85      15,863,040   27             2      248,832  36            15      1,399,680  54            11      684,288   72             2      93,312 108             3      93,312 162             1      20,736 648             1      5,184`

Can you imagine that we could build something like that?
Maq777

Posts: 52
Joined: 30 April 2016