The New Sudoku Players' Forum

[coloin]

Oh Coloin, you are opening a can of worms ( non consecutive worms) I

Cheers tarek im glad it didnt turn out to be a can of worms - but i didnt expect a min clue of 5 [or less]
Surely the number of solution grids wont be that large .... but relabelling is a problem ...
trying to do your 5-puzzle tonight !
C

[Mathimagics]

Non-Consecutive solution grid count …

Nice work!

I wonder if this is the first time that JSudoku has been used to perform a solution grid count for a Sudoku variant?

I have confirmed the counts for the first 2 entries in your original table (167, 220), so I now trust my solver.

For NC+, we can count explicitly, or simply enumerate the NC solutions, testing each one for NC+ property.

Either way, I get these counts (for the case center-square = 1):

Code: Select all

               NC      NC+
25341....     167        4
2.3415...     220       44
2.341..5.    2321      359   
2.341...5     561        0   
253.1.4..    1199       93
2.3.154..     483       66
2.3.1.4.5     822        0
2.3.1..4.   11567     4444
2.3.1...4    1577       19
24..13...    8720     5139
2..413...    1434      135
2...134..   12700      883
2...13.4.   13269     6508
24..1...3   14864     6096
2.4.1...3    3177      389
--------------------------
            73081    24179


[qiuyanzhe]

It turns out that I mistook in calculating permutations, so all my data posted before should be multiplied by 2(edited). Now I think those are correct now, so
#NC=5288648.
And we're done with NC+, total count should be
#NC+=24179*8*9=1740888.
P.S. If all pairs are independent, there should be 6.67e21*(28/36)^144=1280504 NC grids and 6.67e21*(27/36)^144=6807 NC+ ones.. Interesting..

[Mathimagics]

Ok, good job!

Quick question - in your table for NC, why does that last entry have 8, not 16, isotopes?

[qiuyanzhe]

In your table for NC, why does that last entry have 8, not 16, isotopes?

I have to admit that this is not straightforward.. Every grid has 16 isotopes, but only in the last case, solutions are counted twice.

Code: Select all

473.5...6 --(19 28..)->
637.5...4 --(rot 180°)->
4...5.736 --(flip diag)->
4.7.53..6


Also edited minor mistakes in my previous post.

[Mathimagics]

Ok, got it!

To count the number of ED grids for SudokuNC, then, we just need to add up the 1st column of your table, which gives us 344,774, agreed?

This is subject to independent verification, which I am currently doing …

[qiuyanzhe]

It's not.. actually I haven't counted the number of grids with self-symmetry( The only possible symmetry is (rotate 180°+ reverse digits) rxcy + r(10-x)c(10-y)=10 ). We still need to investigate on the number of symmetrical grids..
Suppose it is X, then the solution would be 344,774 - 28667 + (28667-X)/2 + X.
(I bet X would be an odd number! otherwise..there must be something wrong )

[Mathimagics]

Ok, well that will be picked up in my process which is basically:

• Enumerate the solution grids using your CBP list (center box patterns)

• Canonicalise each grid (ie choose min form of 16 possible isotopes)

• Remove duplicates

This will go much faster now that I have got my dobrichev-fsss2 solver working in NC mode …

[tarek]

Re: Isomorphic grids

I got for Non Cyclic NC: 2 (Labelling) x 8 (symmetries) = 16 isomorphs
& for Cyclic NC: 18 (Labelling) x 8 (symmetries) = 144 isomorphs

Is that how you see it?

Here are the 144 isomorphs of a cyclic NC grid for verification

Show 144 cNC isomorphs: Show

Code: Select all

135268497862794135497531862971853624624179358358426971586942713249317586713685249
136497528852631794479258136947582613285316479613974852361749285528163947794825361
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964182537537469281281735964428351796153697428796824153379246815815973642642518379
973815246246379518518642973351428697824796351697153824469537182182964735735281469
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975842613248316975613579248139257486486931752752684139524168397861793524397425861

And the 16 isomorphs if the grid was a non cyclic NC for verification

non cyclic NC 16 isomorphs: Show

Code: Select all

135268497862794135497531862971853624624179358358426971586942713249317586713685249
147926853629358471853174296538417962296835714471692538714269385962583147385741629
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926147583741385269583962417835296174417538692269714835692471358174853926358629741
942586317685713942317249685179624853853971426426358179268135794531497268794862531
963184257481752639257936814572693148814275396639418572396841725148527963725369481
975842613248316975613579248139257486486931752752684139524168397861793524397425861


tarek

[tarek]

Code: Select all

Easy Cyclic NC (8 clues not symmetric)
FP(1,9) redundant but makes solving easier
.....................1...2...8...6...2.............3...7.........2...............
+-------+-------+-------+
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | 1 . . | . 2 . |
+-------+-------+-------+
| . . 8 | . . . | 6 . . |
| . 2 . | . . . | . . . |
| . . . | . . . | 3 . . |
+-------+-------+-------+
| . 7 . | . . . | . . . |
| . . 2 | . . . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+

Easy Cyclic NC (9 clues symmetric)
FP(1,9) needed
........1....2.......................3..1.58..............5........3....6........
+-------+-------+-------+
| . . . | . . . | . . 1 |
| . . . | . 2 . | . . . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . 3 . | . 1 . | 5 8 . |
| . . . | . . . | . . . |
+-------+-------+-------+
| . . . | . 5 . | . . . |
| . . . | . 3 . | . . . |
| 6 . . | . . . | . . . |
+-------+-------+-------+


tarek

[Mathimagics]

I got for Non Cyclic NC: 2 (Labelling) x 8 (symmetries) = 16 isomorphs
& for Cyclic NC: 18 (Labelling) x 8 (symmetries) = 144 isomorphs

Is that how you see it?

Indeed I do!

[tarek]

.
Hmmm, is a one-given puzzle possible? I wonder ...

Possible with the correct set of constraints. IIRC udosuk has posted a puzzle with 1 clue (I can't recall the constraints but i'm expecting there were many). He nicely presented it as a golden egg at the bottom of the empty grid with an Ostrich (Hint) sitting on it. To make the presentation better he broke with convention and used 0 as a clue (Therefore the golden egg)

tarek

[Mathimagics]

Ok, then, we'll have to see what can be done with FP's alone …

Meanwhile, back at the ranch ostrich farm, I haven't been able to post ED counts yet because my fast solver deceived me! It solved a test case correctly, so I trusted it. But when I tested the first of those 9-cell center-box patterns to verify the count, expecting 167, I got 371 (ouch!).

My first Sudoku-variant Solver failure ...


[EDIT] Some consolation, at least it did find all of the 167 valid solutions. And about 500 times faster than my VB6 solver did, so progress of some sort, and perhaps enough to enable a complete enumeration, from which we can produce an ED figure ...

[Mathimagics]

Ok, by my reckoning

• there are 330,845 ED SudokuNC grids

• there are 12,263 ED SudokuNC+ grids

Despite the dodgy solver, it did produce all the valid solutions, just some unwanted rubbish as well. Once we tossed out the trash, and verified the counts, then generated the CF (Canonical Form) for each valid grid, then removed the duplicates, only the ED grids were left.

Full ED grid listings available on request (PM me with an email address).

ED counts are of course provisional, subject to the normal requirement of some independent verification.

[EDIT] Oh dear, made a hash of it. All my CF functions to date have normalised the grid isomorphisms before determining which is smallest lexically, but of course we can't do that here, it totally kills the NC property. Revised figures updated above. (Thanks blue! )

[EDIT #2] Owing to a bug in my NC+ CF function, the NC+ figure above was wrong, the count above now agrees with that reported by blue

[Mathimagics]

... on the number of symmetrical grids ...
Suppose it is X, then the solution would be 344,774 - 28667 + (28667-X)/2 + X.
(I bet X would be an odd number! otherwise..there must be something wrong )

X = 809 seems to be the answer, oddly enough