The New Sudoku Players' Forum

[tarek]

.... I tried to generate puzzles with clues only on the faces of the cube (complete opposite of Barren Surface puzzle) with success ....

That reminds me of Marilyn's Numbrix puzzles in the Sunday Parade newspaper supplement. For the past two years or so, all of hers have clues in, and only in, the odd-numbered squares around the perimeter.

The concept is not that difficult, the Numbrix & Slitherlink are exact cover problems which can be programmed then by choosing the approriate solution grids, generating these grids will be very easy .....

tarek

[tarek]

.... I tried to generate puzzles with clues only on the faces of the cube (complete opposite of Barren Surface puzzle) with success ....

That reminds me of Marilyn's Numbrix puzzles in the Sunday Parade newspaper supplement. For the past two years or so, all of hers have clues in, and only in, the odd-numbered squares around the perimeter.

The concept is not that difficult, the Numbrix & Slitherlink are exact cover problems which can be programmed then by choosing the approriate solution grids, generating these grids will be very easy .....

I have made a mini attempt at a numbrix generator/solver check the Other logic puzzles thread

[coloin]

I enjoyed the challenge of programming this 9*9*9 3D variant. This thread will introduce some concepts and reference material to help solvers of this variant in general and my posted puzzles in particular.

The CUBE

* The complete overlap of 27 Sudoku grids has made it possible to present this variant as a cube.
* There are 6 faces in each cube and 9 grids are positioned in succession spanning the distance between 2 opposite faces.
* This Extreme overlap has made it possible to fit 729 cells (9*9*9) into 27 grids (See picture)



The CELLS and GRIDS

* I will be adopting the following principles when presenting puzzles:

looking horizontally at one surface of the cube, the cells are numbered from 1-729 in a Left to Right, Top to bottom then front to back fashion (See picture)
Cell 1 will always be the leftmost, topmost and frontmost cell - The complete opposite of cell 729 -.
The 27 grids can be presented as 3 ways to slice the cube into 9 equal grids.
9 grids will run from the Front of the cube to the back. 9 will run from top to bottom and 9 grids will run from left to right (This allows an easy correlation between the pictured cube and the grids)
The 9 successive slices (grids) in any of the 3 dimensions will will be sufficient to place the clues in the remaining 18 grids because the grids would have covered all 729 cells.
Presenting the 27 Grids will be preferable for manual solvers.
Programmers and computer solvers would probably like the puzzle in line format (729 characters representing cells 1-729)

* Localizing a cell can be achieved using the cube cell number (1-729) or using the Grid, Row and Column method.

Grids 1-9 will run from the front to the back and can be referenced as g1-g9 or xy1-xy9
Grids 10-18 will run from top to bottom and can be referenced as g10-g18 or xz1-xz9
Grids 19-27 will run from left to right and can be referenced as g19-g27 or yz1-yz9
g1r1c1 is cell 1 g1r9c9 is cell 81
g10r1c1 is cell 649 g10r9c9 is cell 9
g19r1c1 is cell 649 g19r9c9 is cell 73
The pictured cube will help in visualizing this.

I woke up thinking about this last night !!!!!
....
In particular I wondered how many ED ways to fill in the cube in the middle .. cant be many .. and then i wondered how many ways to fill in all 27 solution grids ... and that will be quite a lot !!

[coloin]

the grids (solved puzzles) are isomorphic (?)
not the puzzles
when you know that, you may build that into your solver ...

In particular I wondered how many ED ways to fill in the cube in the middle .. cant be many .. and then i wondered how many ways to fill in all 27 solution grids ...

I struggled to complete a box .... but employing the MC grid i was able to construce ONE

Here are the 6 faces - ready to be folded into a 3*3 cube !

Code: Select all

        +---+     
        |978|     
        |564|     
        |123|     
+---+---+---+---+
|879|951|123|348|
|213|384|456|672|
|546|627|789|915|
+---+---+---+---+
        |789|     
        |231|     
        |645|     
        +---+   

The inside clue in loci {2,2,2] is 9

is this the only way to make a cube i wonder if it really is !
Except in K Pres 's puzzle all the solution grids are essenstially the same [? MC grid]

Code: Select all

256137894489625713371948562137894256625713489948562371894256137713489625562371948
894256137713489625562371948256137894489625713371948562137894256625713489948562371
137894256625713489948562371894256137713489625562371948256137894489625713371948562
562371948894256137713489625371948562256137894489625713948562371137894256625713489
948562371137894256625713489562371948894256137713489625371948562256137894489625713
371948562256137894489625713948562371137894256625713489562371948894256137713489625
713689425542371968896254137689425713371968542254137896425713689968542371137896254
425713689968542371137896254713689425542371968896254137689425713371968542254137896
689425713371968542254137896425713689968542371137896254713689425542371968896254137
689425713425713689713689425371948562948562371562371948137894256894256137256137894
371968542968542371542371968256137894137894256894256137625713489713489625489625713
254137896137896254896254137489625713625713489713489625948562371562371948371948562
425713689713689425689425713948562371562371948371948562894256137256137894137894256
968542371542371968371968542137894256894256137256137894713489625489625713625713489
137896254896254137254137896625713489713489625489625713562371948371948562948562371
713689425689425713425713689562371948371948562948562371256137894137894256894256137
542371968371968542968542371894256137256137894137894256489625713625713489713489625
896254137254137896137896254713489625489625713625713489371948562948562371562371948
647395182395218674218467953476953821953182746182674539764539218539821467821746395
821746395764539218539821467218467953647395182395218674182674539476953821953182746
953182746182674539476953821539821467821746395764539218395218674218467953647395182
476953821953182746182674539764539218539821467821746395647395182395218674218467953
218467953647395182395218674182674539476953821953182746821746395764539218539821467
539821467821746395764539218395218674218467953647395182953182746182674539476953821
764539218539821467821746395647395182395218674218467953476953821953182746182674539
182674539476953821953182746821746395764539218539821467218467953647395182395218674
395218674218467953647395182953182746182674539476953821539821467821746395764539218

So maybe dukuso knew this all along !!!!! And that there really is only one 3D Sudoku Cube !!!!

EDIT
as ever not quite !

Code: Select all

grid solution  1-9
123456789456789123789123456231564897564897231897231564312645978645978312978312645 - MC Grid
grid solution  10-18                                       |  |  |  |     |     |
123456789456789123789123456231564897564897231897231564312648975648975312975312648 - AN OTHER repeating minirow grid - differing only in 6 5<->8 transfers
grid solution  19-27
123456789456789123789123456231564897564897231897231564312648975648975312975312648
 

EDIT the grids and their bands

Code: Select all

123456789456789123789123456231564897564897231897231564312645978645978312978312645   1  1  1 , 1  1  1
123456789456789123789123456231564897564897231897231564312648975648975312975312648   1  1  1 , 2  2  2
123456789456789123789123456231564897564897231897231564312648975648975312975312648   1  1  1 , 2  2  2



hmmmm

[coloin]

On furthur inspection of tarek's puzzle it seems that it doesnt have repeating minirow solution grids...

Ans so maybe it is possible to make a more than one ED 3*3*3* 3D box ... so perhaps things arnt as simple [as usual]

[coloin]

OK ,,,,it seems that dukuso might have been right after all
although Its hard to imagine a set of 9 MC grids ... which map to 2 sets of 9 grids which each have a clue swap in one of the bands ...

Anyway I have been able to make a few more different 3*3*3 boxes...... but unable to work out the ED value

Ive also looked at the solution grids of tarek's #001: The Barren Surface
the 27 grid solutons are composed of 9 lots of 3 ED grids [ 9 in each plane].

Here are the min lex grids followed by the band index

Code: Select all

123456789456789123789123456234567891567891234891234567372615948615948372948372615    1   1   1  ,  30  30  30
123456789456789231789123645238514976514697823697238514362845197845971362971362458   27  27  27  , 310 310 310
123456789456789231789312456231564978564978312978123564312645897645897123897231645   30  30  30  , 413 413 413


I suppose to get a single complete grid to lay over another one and not clash over any of the 81 clues would be improbable ... but doable ... but to repeat this feat 7 more times is pushing it !
....and therefore its probably not surprizing that an isomorph of the same ED grid solution is employed by each plane [9 times]

[tarek]

Hi coloin, It is a shame that this new issue you mention didn't have too many responses. Unfortunately I'm a bit busy lately so will return to this as soon as I have time

tarek

[coloin]

yes.... i suppose looking at a few more puzzle solutions will be informative
There must be some reason why the 3 horizontal or vertical bands have to be the same [essentially].
It would appear that the some of the grids above [1 1 1 , 30 30 30] [30 30 30 , 413 413 413] have an automorphism of 54

presumably it doesnt take long for your program to "solve" the 27 puzzles
It would be a good exercise to get an idea of how many [zillions] or how few [hundreds] of ED 3D cubes there are !! ??

[coloin]

20 [ one 9*9 from each different puzzle] solution grids taken from 20 puzzles from https://www.menneske.no/sudoku3d/eng/

Code: Select all

124678359935412867786593241247986513351724698869135472492861735573249186618357924
152847396634219785978563421561428973347196852289735614715684239423971568896352147
182357649735964218496821573218735496573649821964182357821573964357496182649218735
194537628862419753375286941621394587758162439943875216517623894489751362236948175
239846571568719342147523689614357928973284156852961734785192463421635897396478215
397428651428516397165397428739284516284165739651739284973842165842651973516973842
437912586982536417516487932691758243753241698248693751825364179164879325379125864
471985326593612847268734159859263714126478935347591682632147598784359261915826473
495813627627549813813627954762954138138762495549138762381276549954381276276495381
568219347421735896973684152857961234146523789392478615285197463714356928639842571
627895431895431276431762895314276958762958314958314627589143762143627589276589143
653241798247598613198673245381726459526419387479385126734852961812967534965134872
695187324132564798874293156487329615569718432213456879321645987748932561956871243
719326548682154973435897261978612354263485197541739826397268415126543789854971632
762495138549138762138762954495381276381276549276954381813627495627549813954813627
791832465365794812842165793537429681684517329129683547478251936916378254253946178
839146572472835196156972834724358961561729348398461725615297483983614257247583619
856413729341972685297568134135247968724896513689351472572684391468139257913725846
954762318183954627276831954549276831318549762627183549495627183831495276762318495
963471528285639714147852396396147852528963471714285639639714285852396147471528963


of which only 7 were ED

Code: Select all

963471528285639714147852396396147852528963471714285639639714285852396147471528963 #    1   1   1  ,   1   1   1
194537628862419753375286941621394587758162439943875216517623894489751362236948175 #    1   1   1  ,  30  30  30
695187324132564798874293156487329615569718432213456879321645987748932561956871243 #    1   1   1  , 310 310 310
182357649735964218496821573218735496573649821964182357821573964357496182649218735 #    1   1   1  , 413 413 413
568219347421735896973684152857961234146523789392478615285197463714356928639842571 #   30  30  30  , 413 413 413 *
627895431895431276431762895314276958762958314958314627589143762143627589276589143 #   30  30  30  , 413 413 413 *
239846571568719342147523689614357928973284156852961734785192463421635897396478215 #  310 310 310  , 310 310 310

* - different grids but same bands


am begining to think there are not very many different 3D sudoku grids !!!

And thinking about how many isomorphs is tricky even before you consider the 54 or 648 automorphic reductions
fixing one plane .... 9! * 6^8 * 2
varying the rows and bands in both of the other planes [2^2 * 6^2] ^ 2

Possibly we could define a puzzle solution as the minlex of 9 solution grids of the most min lex plane ... which would be a 9x81 character text string !

[tarek]

am begining to think there are not very many different 3D sudoku grids !!!

And thinking about how many isomorphs is tricky even before you consider the 54 or 648 automorphic reductions
fixing one plane .... 9! * 6^8 * 2
varying the rows and bands in both of the other planes [2^2 * 6^2] ^ 2

Possibly we could define a puzzle solution as the minlex of 9 solution grids of the most min lex plane ... which would be a 9x81 character text string !

dukuso certainly thinks that they are all the same solution grid!!!

[coloin]

Its quite boggling

I cant even work out how many waus to fill a 3x3x3 box !!

Maybe is it true that each plane always has 1-9 labeling isomorphs of the same grid solution ... otherwise how would they overlay each other wthout a constraint ....

[tarek]

I haven't given much thought to this as I share with you the same daunting feeling. I was hoping that dukuso would make a return and expand!!

[coloin]

Certainly I need some sort of assistance ....
I have to say that im inclined to think that there will be a relatively small number of ED Sudoku Cubes [ as opposed to zillions]
Getting any particular solver to interact 27 ways is also not for the fainthearted
The minumum number of clues will be small and the puzzles extraordinarliy hard - as you have alluded to !