## SudokuPW (SudokuP + Windoku)

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Re: SudokuPW (SudokuP + Windoku)

Relabelling is pattern independent
36+1 ways of relabelling 9 position pattern with a minimum 8 different clues:
All 9 clues 1 way
8 clues with 1 repeated clue value: 1:8 ways+2:7 ways+3:6 ways+4:5 ways+ ... + 8:1way = 36 ways

When we have symmetric puzzles there could be methods to reduce the need to check all 36 ways because you can predict theat some of the re-labelling is isomorphic to re-labelling that was done earlier!

The following are 2 SPW puzzles midway through our re-labelling process
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`..1..2...........3........4.............4.............6........7...........8..9....1..2...........3........4.............5.............5........7...........8..9..`

You can see that they are isomorphic … Relabelling with 5: 4 ways,6: 3 ways,7: 2 ways and 8: 1 way is not needed as they are isomorphic through labelling. That cuts down the number of re-labelling by 10 to 26 ways

With the Extremely fast solver you are using it may make a very small difference. Using my solvers & processor, I have to cut as many corners as possible!

Tarek

tarek

Posts: 2792
Joined: 05 January 2006

### Re: SudokuPW (SudokuP + Windoku)

Mathimagics wrote:Thus, to enumerate 8-clue puzzles, I simply select all possible sets of 4 cells in the first 4 rows (58905 ways to pick 4 from 36), and complete each pattern by adding the 4 "opposite" cells.

9-clue puzzle patterns are just the 8-clue patterns with cell (5, 5) added.

You need to choose 4 of 40 cell pairs (not 36), or you'll miss puzzles like this:

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`. . . . . . . . .. . . . . . . . .. . . . 1 . . 2 .. . . 3 . . . . .. . 4 . 5 . 6 . .. . . . . 2 . . .. 7 . . 8 . . . .. . . . . . . . .. . . . . . . . .PWX: (8-digit) 9-clue`
blue

Posts: 740
Joined: 11 March 2013

### Re: SudokuPW (SudokuP + Windoku)

I wonder if that 36 was in reference to the 36 ways you can re-label 8 different clues in 9 clue positions. My guess that it was a typo or it will also risk missing this PWX
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`+-------+-------+-------+| 1 . . | . . . | . . . || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| . . 2 | . . . | . . . || . 3 . | 2 4 5 | . 6 . || . . . | . . . | 7 . . |+-------+-------+-------+| . . . | . . . | . . . || . . . | . . . | . . . || . . . | . . . | . . 8 |+-------+-------+-------+`

tarek

tarek

Posts: 2792
Joined: 05 January 2006

### Re: SudokuPW (SudokuP + Windoku)

.
Ok, thanks tarek, now I get it!

And thank you blue for pointing out my pattern omissions. Sloppy work on my part, once again!

tarek wrote:I wonder if that 36 was in reference to the 36 ways you can re-label 8 different clues in 9 clue positions. My guess that it was a typo or it will also risk missing this PWX ...

No, it was a genuine error on my part, I was building 8C/9C patterns by first choosing 4 cells from the first 4 rows (36 cells) and reflecting them to get 8C. I totally missed the constructions involving clues in R5.

The total base pattern count for 8C/9C patterns is indeed Combns(40, 4) = 91390.

And for 10C/11C the count should be Combns(40, 5) = 658008.

Mathimagics
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### Re: SudokuPW (SudokuP + Windoku)

.
Ok, the updated symmetric puzzle counts are:

• SudokuPW: 8C = n/a, 9C = 64
• SudokuPWX: 8C = 8, 9C = 7696

Only the 9C PWX count has changed, and I checked that the new list includes both tarek's and blue's missing cases.

Mathimagics
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### Re: Double Diagonal 9 clue SudokuPW (Not for manual solving)

From my last run:

There were a few double diagonal symmetric Sudoku PW (They might turn out to be the same puzzle). All Toughies & I suspect that they are all the same puzzle

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

One of the puzzles image
Hidden Text: Show

tarek
Last edited by tarek on Thu Dec 13, 2018 1:17 am, edited 2 times in total.

tarek

Posts: 2792
Joined: 05 January 2006

### Re: Manually solvable symmetric 9 clue Sudoku PW

On a better note I managed to find some solvable Sudoku PW & here are 2

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`Sudoku PW (Disjoint Groups + Windoku) Easy but beyond singles........1..........2...........................13.......45.....6.....7...8.......+-------+-------+-------+| . . . | . . . | . . 1 || . . . | . . . | . . . || . 2 . | . . . | . . . |+-------+-------+-------+| . . . | . . . | . . . || . . . | . . . | . . . || . . 1 | 3 . . | . . . |+-------+-------+-------+| . . 4 | 5 . . | . . . || 6 . . | . . . | 7 . . || . 8 . | . . . | . . . |+-------+-------+-------+`

Hidden Text: Show

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`Sudoku PW (Disjoint Groups + Windoku) Advanced fish......1..2........................2.34.5............6..........7..............8..+-------+-------+-------+| . . . | . . . | 1 . . || 2 . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| . . . | . . . | . 2 . || 3 4 . | 5 . . | . . . || . . . | . . . | . 6 . |+-------+-------+-------+| . . . | . . . | . . . || 7 . . | . . . | . . . || . . . | . . . | 8 . . |+-------+-------+-------+`

Hidden Text: Show

tarek

tarek

Posts: 2792
Joined: 05 January 2006

### Re: Double Diagonal 9 clue SudokuPW (Not for manual solving)

tarek wrote: All Toughies & I suspect that they are all the same puzzle
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`..1.........2.....3.........4...........5...........1.........6.....7.........8...1.......2...........3.......4..........5..........1.......6...........7.......8.`

Confirmed, these two have the same CF (canonical form).

Mathimagics
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Joined: 27 May 2015

### 4-fold symmetric 9 clue PWX

Also some nice 4-fold symmetry 9 clue PWXs .... All here are not aimed at Manual solving

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`90 degrees rotational..1..2....................34............3............56....................7..8..+-------+-------+-------+| . . 1 | . . 2 | . . . || . . . | . . . | . . . || . . . | . . . | . . 3 |+-------+-------+-------+| 4 . . | . . . | . . . || . . . | . 3 . | . . . || . . . | . . . | . . 5 |+-------+-------+-------+| 6 . . | . . . | . . . || . . . | . . . | . . . || . . . | 7 . . | 8 . . |+-------+-------+-------+Double diagonal.1.......2...........2.......3..........4..........5.......6...........7.......8.+-------+-------+-------+| . 1 . | . . . | . . . || 2 . . | . . . | . . . || . . . | 2 . . | . . . |+-------+-------+-------+| . . 3 | . . . | . . . || . . . | . 4 . | . . . || . . . | . . . | 5 . . |+-------+-------+-------+| . . . | . . 6 | . . . || . . . | . . . | . . 7 || . . . | . . . | . 8 . |+-------+-------+-------+Double orthogonal.............1.....2.....3.............345.............6.....7.....8.............+-------+-------+-------+| . . . | . . . | . . . || . . . | . 1 . | . . . || . 2 . | . . . | . 3 . |+-------+-------+-------+| . . . | . . . | . . . || . . . | 3 4 5 | . . . || . . . | . . . | . . . |+-------+-------+-------+| . 6 . | . . . | . 7 . || . . . | . 8 . | . . . || . . . | . . . | . . . |+-------+-------+-------+`

Hidden Text: Show

Mathimagics wrote:Confirmed, these two have the same CF (canonical form).

Thanks for the confirmation

tarek

Posts: 2792
Joined: 05 January 2006

### Re: Solvable Symmetric 9 clue PWX

There should be more but here are 2

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`Sudoku PWX (Disjoint Groups + Windoku + X) Difficult: advanced fish.......1...2......................3.45..6...........2............7.............8.+-------+-------+-------+| . . . | . . . | . 1 . || . . 2 | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| . . . | . . . | . 3 . || 4 5 . | . 6 . | . . . || . . . | . . . | . 2 . |+-------+-------+-------+| . . . | . . . | . . . || . . 7 | . . . | . . . || . . . | . . . | . 8 . |+-------+-------+-------+`

Hidden Text: Show

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`Sudoku PWX (Disjoint Groups + Windoku + X) Easy but beyond singles.......1..23.....4.....2........56...........................7........8..........+-------+-------+-------+| . . . | . . . | . 1 . || . 2 3 | . . . | . . 4 || . . . | . . 2 | . . . |+-------+-------+-------+| . . . | . . 5 | 6 . . || . . . | . . . | . . . || . . . | . . . | . . . |+-------+-------+-------+| . . . | . . . | . 7 . || . . . | . . . | . 8 . || . . . | . . . | . . . |+-------+-------+-------+`

Hidden Text: Show

tarek

tarek

Posts: 2792
Joined: 05 January 2006

### Re: SudokuPW (SudokuP + Windoku)

.
I forgot to post the final tallies of 8-clue puzzles:

• SudokuPW: ED grids = 177564, 8C puzzles = 208 (77 grids, 0.04%)
• SudokuPWX: ED grids = 2922, 8C puzzles = 59282 (1067 grids, 36.5%)

Mathimagics
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Joined: 27 May 2015

### Re: SudokuPW (SudokuP + Windoku)

Code: Select all
`PW :   190 ED puzzles,   77 ED gridsPWX: 55841 ED puzzles, 1069 ED grids`

Hopfully, your "1067" number for PWX grids, was just a typo.
If you post zip files of the 208 PW puzzles, and 1067 PWX grids, I'll compare the results against mine.

The method I used was:
• Enumerate the (81! / 73! / 8!) eight-cell patterns.
• Ignore patterns that are not minlex w/rsp to the 16 VPTs
• For the rest ... 2,011,185,998 ED patterns in all: Test the corresponding 8-clue puzzles for validity.
• Output the valid puzzles.
Note: The puzzles too, are minlex w/rsp to the 16 VPTs; and they're distinct, and so they're collectively "ED".
• In a separate pass: Re-solve the puzzles, canonicalize the solution grids, and output 1st occurences for the resulting grids.

Added: The different results are partially explained (on my end), by applying solution grid automorphisms to puzzles with "automorphic" solution grids. It looks like the "1067" wasn't a typo

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`PW :   190 ED puzzles,   77 ED grids :   208 puzzles total, on the   77 ED (solution) gridsPWX: 55841 ED puzzles, 1069 ED grids : 59318 puzzles total, on the 1069 ED (solution) grids`
blue

Posts: 740
Joined: 11 March 2013

### Re: SudokuPW (SudokuP + Windoku)

.
Hi blue!

My puzzle counts were Hitting-set method generation on each ED grid, counting all distinct 8-clue puzzles. No puzzle-filtering.

So for PW we seem to coincide, I take it?

For PWX I think the job may have missed a couple of grids. This is entirely possible as the job had to be restarted a couple of times (thunderstorm cut power, and destroyed my TV/Monitor, plus a mysterious unknown cause reboot). Your total puzzles count exceeds mine by just 36, so the 2-grid difference would make sense.

Perhaps I could just give you a list of the PWX grid's for which I found puzzles? Will that suffice? Hopefully our CF's match up ... in which case I could probably just give you a list of the CF indices (grid #'s)

Mathimagics
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Joined: 27 May 2015

### Re: SudokuPW (SudokuP + Windoku)

Hi Mathimagics,

PW results match, yes.
Sorry to hear about the power outage, and so on

Full grids would be quickest for me to handle ... just a few minutes.
CF indicies would take a little longer.
blue

Posts: 740
Joined: 11 March 2013

### Re: SudokuPW (SudokuP + Windoku)

.
Ok, done. List of 1067 grids attached.
Attachments
SudokuPWX-8CG.txt