The New Sudoku Players' Forum

[tarek]

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

[blue]

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

[tarek]

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

[Mathimagics]

.
Ok, thanks tarek, now I get it!

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

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]

.
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.

[tarek]

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

[tarek]

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

Code: Select all

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

[Mathimagics]

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).

[tarek]

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

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

Thanks for the confirmation

[tarek]

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

Code: Select all

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

[Mathimagics]

.
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%)

[blue]

I had slightly different results:

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PW :   190 ED puzzles,   77 ED grids
PWX: 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

Code: Select all

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

[Mathimagics]

.
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)

[blue]

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.

[Mathimagics]

.
Ok, done. List of 1067 grids attached.