The New Sudoku Players' Forum

[Mathimagics]

.
A SudokuPX grid has all of the properties of a SudokuP grid, plus diffent values along both diagonals.

I counted these some time ago, but never got around to posting the results.

These grids have far fewer PX-preserving transformations (morph options), in fact as far as I can tell we are reduced to the normal dihedral symmetries, namely transpose and rotation, so there are only 8 morph flavours, none of any interest. Perhaps blue can uncover some!

• there are 133,747,300 different SudokuPX grids (up to relabelling the digits)

• for PX-equivalence under the 8 PX-preserving transformations, we simply enumerated them explicitly and determined the number of PX-different grids as 16,724,358

• for S-equivalence, ie essentially different in the normal Sudoku-equivalence sense, we don't really know. I will leave the ED count calculation as an exercise for Serg

The main question here is the minimum number of clues. We know 12-clue puzzles have been found for SudokuX so the chances are good that we'll find 10-clue, maybe even 8-clue SudokuPX puzzles.

Given the low numbers of grids involved, and that my HS engine is currently in pieces on the garage floor, I invite blue and champagne to perhaps provide an answer to this question.

[blue]

These grids have far fewer PX-preserving transformations (morph options), in fact as far as I can tell we are reduced to the normal dihedral symmetries, namely transpose and rotation, so there are only 8 morph flavours, none of any interest. Perhaps blue can uncover some!

There are more ... 32 in all.

• there are 133,747,300 different SudokuPX grids (up to relabelling the digits)

• for PX-equivalence under the 8 PX-preserving transformations, we simply enumerated them explicitly and determined the number of PX-different grids as 16,724,358

I can confirm both of those numbers.
For the 2nd one, when all 32 transformations are employed, the ED/"PX-different" count drops to 4,208,450.

For the missing transformations:

• there's a factor of 2, from the transformation that swaps bands 1&3 and stacks 1&3, or alternatively from the transformation that swaps rows 1&3, 4&6, 7&9, and columns 1&3, 4&6, 7&9.
• there's a factor of 2 from the "E" transformation -- swapping rows 2&4, 3&7, 6&8, and columns 2&4, 3&7, 6&8.

[Mathimagics]

.
Nice one, blue, thanks!

I will retrofit your additional transformations to my catalog ...

[Leren]

A couple of early results for SudokuPX:

1. B8 is not only an 11 clue puzzle for SudokuP it is also an 11 clue puzzle for SudokuPX :

......7.9.......3..6...2............6..............8.........1........4.7...8.... 123456789457891236968372451571948362689523174234617895895234617312765948746189523

2. I've found three 10 clue puzzles for SudokuPX :

1.2........3..............4.4.......56..7...........2..8......................... 192564738453817296876293154249356817568172349731489625985621473624735981317948562
1.2........3..............4.4..5.....6..............1..7...........8............. 182435697493678251756291384241359768867124539539867412974516823315782946628943175
1.2........3..4...........5.5..6.....7..............1..8......................... 142536798593784261867291345251369874478125639639478512985617423316842957724953186

The trick to all this is to use the 11 clue SudokuP puzzles as a likely source of good things in SodokuXP. This intuition has born fruit :

The first puzzle was found by noting that O2 minus it's last clue 1.2........3..............4.4..5.....6..7...........2..8......................... has exactly 2 SudokuXP solutions :

192534678453687192876192354349251867268379541517468923785923416924816735631745289
192564738453817296876293154249356817568172349731489625985621473624735981317948562

From this information numerous 11 clue SudokuXP puzzles can be found, with one 10 clue puzzle in the mix. The second 10 clue puzzle is just O1 minus its last clue and the third is O3 minus its last clue.

Leren

[Mathimagics]

.
Confirmed!

Well done Leren!

Code: Select all

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


[Leren]

A quick run through all 14 11 clue SudokuP puzzles has revealed 2 more 10 clue SudokuPX puzzles :

.2.45..........13........6.....9...........7............7..............5......... 123456789756289134984731562375694218641528973298173456537962841462817395819345627 B1 minus last clue
.....6.....7.....2..........................3..5..........3.....8....96..4....... 294386571657149832138752496813427659426598713975613284762935148381274965549861327 B7 minus 2nd Clue

PS - I've also quickly checked the F, G and E transforms of the 11 clue SudokuP puzzles, and it appears that E(X) is ESPX (X) but F(X) and G(X) are EDPX(X).

eg E(O2) minus one of its clues has exactly 2 SudokuPX solutions and by exchanging corresponding clues as in the untransformed case you get a 10 clue puzzle as for O2 with exactly the same solution signature, so it's probably a PX morph of the first 10 clue puzzle.

The result is 1.....2.....4........8...........3..5..67.....................4.....2............ 157963248238451967964827513482519376513674829679238451821795634746382195395146782

Leren

[Leren]

Found 1 more 10 clue SudokuPX puzzle, to bring the total to 7, although I suspect no 6 is ESPX to no 1. Here they all are, with an indication as to how I found them.

1.2........3..............4.4.......56..7...........2..8......................... 192564738453817296876293154249356817568172349731489625985621473624735981317948562 Variation of O2
1.2........3..4...........5.5..6.....7..............1..8......................... 142536798593784261867291345251369874478125639639478512985617423316842957724953186 O3 minus last clue
1.2........3..............4.4..5.....6..............1..7...........8............. 182435697493678251756291384241359768867124539539867412974516823315782946628943175 O1 minus last clue
.2.45..........13........6.....9...........7............7..............5......... 123456789756289134984731562375694218641528973298173456537962841462817395819345627 B1 minus last clue
.....6.....7.....2..........................3..5..........3.....8....96..4....... 294386571657149832138752496813427659426598713975613284762935148381274965549861327 B7 minus 2nd Clue (4)
1.....2.....4........8...........3..5..67.....................4.....2............ 157963248238451967964827513482519376513674829679238451821795634746382195395146782 Variation of E(O2)
.2.......45..........13..6...........9...........7......7....................5... 123756984456289731789134562375641298694528173218973456537462819962817345841395627 B2 minus 2nd last clue

So where to from here for this project ? Do we institute an exhaustive search for more 10's, with the possibility of finding 9, or even fewer, clue puzzles ?

Leren

[Mathimagics]

.
For SudokuP we see many thousands of 12-clue puzzles, but only tiny numer of 11-clue puzzles.

I've been running the Morph Walker (the ghost who walks!) in SudokuPX mode. It has found 2000 11-clue puzzles so far, but every one is minimal.

This suggests that we might have a similar scenario, namely that 10-clues is the limit, and that there are not very many of them.

In which case Leren's 10-clue finds are all the more remarkable.

So I think we need to do a rigorous HS search for 10-clue puzzles, which would also prove/disprove non-existence of 9-clue puzzles.

[Leren]

I also found many minimal 11's when looking for the 10' s. Perhaps someone would be kind enough to convert my 10's results into minlex format, similar to what was done for SudokuP.

This should also confirm whether or not results 1 and 6 are ES or not. I suspect that they are.

Leren

[Mathimagics]

.

I don't have a SudokuPX CF function, so I compared the grids with the SudokuP CF function:

Code: Select all

192564738453817296876293154249356817568172349731489625985621473624735981317948562 # 23050009
142536798593784261867291345251369874478125639639478512985617423316842957724953186 # 20226364
182435697493678251756291384241359768867124539539867412974516823315782946628943175 # 20226364
123456789756289134984731562375694218641528973298173456537962841462817395819345627 # 15264367
294386571657149832138752496813427659426598713975613284762935148381274965549861327 # 53535252
157963248238451967964827513482519376513674829679238451821795634746382195395146782 # 23050009
123756984456289731789134562375641298694528173218973456537462819962817345841395627 # 15264367


So 1 and 6 are the same, same for (2 and 3), (4 and 7)

[blue]

Here are the (PX-)canonical forms for Leren's puzzles and thier solutions.

(1) and (6) are identical, as predicted.
(2) and (3) have PX-isomorphic solutions.
(4) and (7) are PX-unrelated.

Code: Select all

Minlex puzzle forms:

.........................1..2...........3..45.......6.6..............7........2.8  Variation of O2
.........................1..2..............3.....4..5.5...........6..7........8.2  O3 minus last clue
........................1...1............2.3.....4..565....................7..8..  O1 minus last clue
.......................1...........2.....3.....4..5.........3......6.....7.82....  B1 minus last clue
.......................1........2..3..4...........5......6...7.5...........84....  B7 minus 2nd Clue (4)
.........................1..2...........3..45.......6.6..............7........2.8  Variation of E(O2)
.................1.................2.....3..4..5.............6.....7.85.3........  B2 minus 2nd last clue

Minlex (solution,puzzle) forms:

123456789459718326867293154942375861635184297718629435294537618571862943386941572
..3..............6......1.4.4...........8.29........3..........................7.  Variation of O2

123456789468917253975283641547391862681725394392864517214678935839542176756139428
1...........................47.9.....8..............1.2..6........5...........4..  O3 minus last clue

123456789468917253975283641547391862681725394392864517214678935839542176756139428
1...........................47.9.....8...5..........1.2..6....................4..  O1 minus last clue

123456789456789321798213456671945832245138697389627145567394218914872563832561974
.....6..........................5..2.....8.....9.........39..1.....7....8........  B1 minus last clue

123456789486972531597381624864297315715643298932815476671538942359124867248769153
....5....48.....3........2............5..............6.....8.....9.....7.........  B7 minus 2nd Clue (4)

123456789459718326867293154942375861635184297718629435294537618571862943386941572
..3..............6......1.4.4...........8.29........3..........................7.  Variation of E(O2)

123456789456798213798321456671245398845139627932687145567814932384972561219563874
...........6.......................8..5..9.....2............9..38..7.....1.......  B2 minus 2nd last clue

[Leren]

OK, so far we have 6 EDPX puzzles solving to 5 EDPX grids. I've now automated the 10 clue puzzle search process on a pattern common to many of the known puzzles.

Here is a list of all 82 apparently different 10 clue SudokuPX puzzles from that search. Whether the puzzles and/or grids they solve to are EDPX from the others, or there are any errors, I'll leave to the minlexing experts to determine.

Hidden Text: Show

Seq Puzzle Solution
1 : .....6.....7.....2..........................3..5..........3.....8....96..4....... 294386571657149832138752496813427659426598713975613284762935148381274965549861327
2 : .2.......45..........13..6...........9...........7......7....................5... 123756984456289731789134562375641298694528173218973456537462819962817345841395627
3 : .2.45..........13........6.....9...........7............7..............5......... 123456789756289134984731562375694218641528973298173456537962841462817395819345627
4 : 1.....2.....4........8...........3..5..67.....................4.....2............ 157963248238451967964827513482519376513674829679238451821795634746382195395146782
5 : 1.2........3................4..7.....6...........2.....8.....5.............7..... 192468537653197482478253691845671923269835714317924865784319256921546378536782149
6 : 1.2........3..............4.4........6...............6.8...........7...........5. 172498563453716982698523174849635721367241895215987346986354217524179638731862459
7 : 1.2........3..............4.4........6...............6.8..7....................5. 192438765453796182678125394349651827865247931217983546986574213524319678731862459
8 : 1.2........3..............4.4........6...............6.8..7....................9. 152438769493756182678129354345691827869247531217583946586974213924315678731862495
9 : 1.2........3..............4.4........6.............72..8........................5 172534968493786512856192374347219856268375149519468723785623491924851637631947285
10 : 1.2........3..............4.4........6.............72..8........................9 172934568453786912896152374347215896268379145915468723789623451524891637631547289
11 : 1.2........3..............4.4........6.............73..8.....5................... 172894365493156827856372194748239516365781249219465738987623451534918672621547983
12 : 1.2........3..............4.4........6.............73..8.....9................... 172854369453196827896372154748235916369781245215469738587623491934518672621947583
13 : 1.2........3..............4.4........6.............92..8........................5 192534768473986512856172394349217856268395147517468923985623471724851639631749285
14 : 1.2........3..............4.4........6.............92..8........................7 192734568453986712876152394349215876268397145715468923987623451524871639631549287
15 : 1.2........3..............4.4........6.............93..8.....5................... 192874365473156829856392174948237516365981247217465938789623451534718692621549783
16 : 1.2........3..............4.4........6.............93..8.....7................... 192854367453176829876392154948235716367981245215467938589623471734518692621749583
17 : 1.2........3..............4.4........6..7...........2..8..5...................... 172435698493687152856192374347821569269574831518963427984756213725318946631249785
18 : 1.2........3..............4.4........6..7...........2..8..9...................... 172439658453687192896152374347821965265974831918563427584796213729318546631245789
19 : 1.2........3..............4.4........6.5...........72..8......................... 192754863453816972876392154348279516267581349519463728985627431724138695631945287
20 : 1.2........3..............4.4........6.5...........92..8......................... 172954863453816792896372154348297516269581347517463928785629431924138675631745289
21 : 1.2........3..............4.4........6.7...........92..8......................... 152974863473816592896352174348295716269781345715463928587629431924138657631547289
22 : 1.2........3..............4.4........6.9...........72..8......................... 152794863493816572876352194348275916267981345915463728589627431724138659631549287
23 : 1.2........3..............4.4.......56..7...........2..8......................... 192564738453817296876293154249356817568172349731489625985621473624735981317948562
24 : 1.2........3..............4.4.......96..7...........2..8......................... 152964738493817256876253194245396817968172345731485629589621473624739581317548962
25 : 1.2........3..............4.4..5.....6..............1..7...........8............. 182435697493678251756291384241359768867124539539867412974516823315782946628943175
26 : 1.2........3..............6.4........6..............2..8...........79............ 192756834653184792478392156549218367267935418831647925985421673326879541714563289
27 : 1.2........3..............6.4........6..............2..8...........97............ 172956834653184972498372156547218369269735418831649725785421693326897541914563287
28 : 1.2........3..............6.4........6.....5...........8......................73. 172564398693817542854392176947256813368471259521983467485739621736125984219648735
29 : 1.2........3..............6.4........6.....5...........8......................93. 192564378673819542854372196749256813368491257521783469485937621936125784217648935
30 : 1.2........3..............6.4........6.....7...........8......................93. 192764358653819742874352196549276813368491275721583469487935621936127584215648937
31 : 1.2........3..............6.4........6.....9...........8......................73. 172964358653817942894352176547296813368471295921583467489735621736129584215648739
32 : 1.2........3..............8.4........6.............53..8......7.................. 172968354893514672654372198745639821368251749921847536586493217437126985219785463
33 : 1.2........3..............8.4........6.............53..8......9.................. 192768354873514692654392178945637821368251947721849536586473219439126785217985463
34 : 1.2........3..............8.4........6.............73..8......5.................. 152968374893714652674352198547639821368271549921845736786493215435126987219587463
35 : 1.2........3..............8.4........6.............73..8......9.................. 192568374853714692674392158947635821368271945521849736786453219439126587215987463
36 : 1.2........3..............8.4........6.............93..8......5.................. 152768394873914652694352178549637821368291547721845936986473215435126789217589463
37 : 1.2........3..............8.4........6.............93..8......7.................. 172568394853914672694372158749635821368291745521847936986453217437126589215789463
38 : 1.2........3...........4....4........6..........72.....8.....5................... 172893465453176892698254173547618329269345718831729546786931254924587631315462987
39 : 1.2........3...........4....4........6..........72.....8.....9................... 172853469493176852658294173947618325265349718831725946786531294524987631319462587
40 : 1.2........3...........4....4........6..........72.5...8......................... 172893465453176892698254173547618329269345718831729546786931254924587631315462987
41 : 1.2........3...........4....4........6..........72.9...8......................... 172853469493176852658294173947618325265349718831725946786531294524987631319462587
42 : 1.2........3...........4....4........6..........92.....8.....5................... 192873465453196872678254193549618327267345918831927546986731254724589631315462789
43 : 1.2........3...........4....4........6..........92.....8.....7................... 192853467473196852658274193749618325265347918831925746986531274524789631317462589
44 : 1.2........3...........4....4........6..........92.5...8......................... 192873465453196872678254193549618327267345918831927546986731254724589631315462789
45 : 1.2........3...........4....4........6..........92.7...8......................... 192853467473196852658274193749618325265347918831925746986531274524789631317462589
46 : 1.2........3...7............4........6...........2.....8.....5.............8..... 172938564853146792496257381349715826267389415518624973681493257924571638735862149
47 : 1.2........3...9............4........6...........2.....8.....5.............8..... 192738564853146972476259381347915826269387415518624793681473259724591638935862147
48 : 1.2........3..4...........5.5..6.....7..............1..8......................... 142536798593784261867291345251369874478125639639478512985617423316842957724953186
49 : 1.2........3..5.............4........6..............1..8......6.....7............ 152698437673145982894372651741923865368751294925486713589214376416837529237569148
50 : 1.2........35...............4........6..........3......8..............72......... 152463897693587241874912653941278536368159724527346189789624315416835972235791468
51 : 1.2........35...............4........6..........3......8..............92......... 152463879673589241894712653741298536368157924529346187987624315416835792235971468
52 : 1.2........37...............4........6..........3......8..............92......... 172463859653789241894512673541298736368175924729346185985624317416837592237951468
53 : 1.2........39...............4........6..........3......8..............72......... 192463857653987241874512693541278936368195724927346185785624319416839572239751468
54 : 1.2.....5..3...........4....4........6..........72.....8......................... 172893465453176892698254173547618329269345718831729546786931254924587631315462987
55 : 1.2.....5..3...........4....4........6..........92.....8......................... 192873465453196872678254193549618327267345918831927546986731254724589631315462789
56 : 1.2.....7..3................4........6...........2.....8.....5.............7..... 192468537653197482478253691845671923269835714317924865784319256921546378536782149
57 : 1.2.....7..3...........4....4........6..........92.....8......................... 192853467473196852658274193749618325265347918831925746986531274524789631317462589
58 : 1.2.....9..3...........4....4........6..........72.....8......................... 172853469493176852658294173947618325265349718831725946786531294524987631319462587
59 : 1.2.....9..3...........6....4........6..........72.....8......................... 172834569653972814498156372945618723267345198831729645386497251729581436514263987
60 : 1.2...5....3................4........6.................8.....7.............81.... 192738564653942187478165293847623915365491728921587346286359471514276839739814652
61 : 1.2...5....3................4........6.................8.....9.............81.... 172938564653742189498165273849623715365471928721589346286357491514296837937814652
62 : 1.2...7....3................4........6.................8.....9.............81.... 152938764673542189498167253849623517367451928521789346286375491714296835935814672
63 : 1.2...9....3................4........6.................8.....5.............81.... 172538964693742185458169273845623719369471528721985346286397451914256837537814692
64 : 1.2...9....3................4........6.................8.....7.............81.... 152738964693542187478169253847623519369451728521987346286395471914276835735814692
65 : 1.2..5.....3..............4.4........6.................8.....................6.7. 172495836453681792698732154547218369869374215321569487784953621936127548215846973
66 : 1.2..5.....3..............4.4........6..7...........2..8......................... 172435698493687152856192374347821569269574831518963427984756213725318946631249785
67 : 1.2..5.....3..............6.4..7.....6.................8.......................1. 192765438673824591854391276541673829268159347937482165386217954415936782729548613
68 : 1.2..5.....3..............6.4..7.....6..............1..8......................... 192765438673824591854391276541673829268159347937482615386217954415936782729548163
69 : 1.2..5.....3..............6.4.8......6.................8..7...................... 192465738653287941874913256941856327765132894328749165289574613416328579537691482
70 : 1.2..5.....3..............6.4.8......6.................8..9...................... 172465938653289741894713256741856329965132874328947165287594613416328597539671482
71 : 1.2..7.....3..............6.4.7......6.................8..9...................... 152867934693412785874359126348726519269531847517948263485293671926174358731685492
72 : 1.2..7.....3..............6.4.8......6.................8..5...................... 192467538673285941854913276941876325567132894328549167289754613416328759735691482
73 : 1.2..7.....3..............6.4.8......6.................8..9...................... 152467938673289541894513276541876329967132854328945167285794613416328795739651482
74 : 1.2..9.....3................4..5.....6.............92..8......................... 172539864653842179498671352947258613261397485835164927386425791724916538519783246
75 : 1.2..9.....3................4..7.....6.............92..8......................... 152739864673842159498651372945278613261395487837164925386427591524916738719583246
76 : 1.2..9.....3..............4.4........6.................8.....................6.7. 172459836493681752658732194947218365865374219321965487784593621536127948219846573
77 : 1.2..9.....3..............4.4........6..7...........2..8......................... 172439658453687192896152374347821965265974831918563427584796213729318546631245789
78 : 1.2..9.....3..............6.4.8......6.................8..5...................... 172469538693285741854713296741896325569132874328547169287954613416328957935671482
79 : 1.2..9.....3..............6.4.8......6.................8..7...................... 152469738693287541874513296541896327769132854328745169285974613416328975937651482
80 : 1.2..9.....3..............6.4.9......6.................8..7...................... 152869734673412985894357126348926517267531849519748263485273691726194358931685472
81 : 1.29.......3................4........6..............1..8..5.....................8 172934865853672149694185273548213697261597384937846512386459721415728936729361458
82 : 1.29.......3..............8.4.5......6..............1..8......................... 152978463873426951496351278241537896568149732937862514789614325314295687625783149

Leren

<Edit> I've examined the solution path of all the above puzzles to determine the number that are ED. I think the number is 34 and I've included an example of each in the following list. The minlexers could examine both lists to see if I'm right.

Hidden Text: Show

1 : .....6.....7.....2..........................3..5..........3.....8....96..4....... 294386571657149832138752496813427659426598713975613284762935148381274965549861327
2 : .2.......45..........13..6...........9...........7......7....................5... 123756984456289731789134562375641298694528173218973456537462819962817345841395627
3 : .2.45..........13........6.....9...........7............7..............5......... 123456789756289134984731562375694218641528973298173456537962841462817395819345627
4 : 1.....2.....4........8...........3..5..67.....................4.....2............ 157963248238451967964827513482519376513674829679238451821795634746382195395146782
5 : 1.2........3................4..7.....6...........2.....8.....5.............7..... 192468537653197482478253691845671923269835714317924865784319256921546378536782149
6 : 1.2........3..............4.4........6...............6.8...........7...........5. 172498563453716982698523174849635721367241895215987346986354217524179638731862459
7 : 1.2........3..............4.4........6...............6.8..7....................5. 192438765453796182678125394349651827865247931217983546986574213524319678731862459
9 : 1.2........3..............4.4........6.............72..8........................5 172534968493786512856192374347219856268375149519468723785623491924851637631947285
11 : 1.2........3..............4.4........6.............73..8.....5................... 172894365493156827856372194748239516365781249219465738987623451534918672621547983
17 : 1.2........3..............4.4........6..7...........2..8..5...................... 172435698493687152856192374347821569269574831518963427984756213725318946631249785
19 : 1.2........3..............4.4........6.5...........72..8......................... 192754863453816972876392154348279516267581349519463728985627431724138695631945287
25 : 1.2........3..............4.4..5.....6..............1..7...........8............. 182435697493678251756291384241359768867124539539867412974516823315782946628943175
26 : 1.2........3..............6.4........6..............2..8...........79............ 192756834653184792478392156549218367267935418831647925985421673326879541714563289
28 : 1.2........3..............6.4........6.....5...........8......................73. 172564398693817542854392176947256813368471259521983467485739621736125984219648735
32 : 1.2........3..............8.4........6.............53..8......7.................. 172968354893514672654372198745639821368251749921847536586493217437126985219785463
38 : 1.2........3...........4....4........6..........72.....8.....5................... 172893465453176892698254173547618329269345718831729546786931254924587631315462987
40 : 1.2........3...........4....4........6..........72.5...8......................... 172893465453176892698254173547618329269345718831729546786931254924587631315462987
46 : 1.2........3...7............4........6...........2.....8.....5.............8..... 172938564853146792496257381349715826267389415518624973681493257924571638735862149
48 : 1.2........3..4...........5.5..6.....7..............1..8......................... 142536798593784261867291345251369874478125639639478512985617423316842957724953186
49 : 1.2........3..5.............4........6..............1..8......6.....7............ 152698437673145982894372651741923865368751294925486713589214376416837529237569148
50 : 1.2........35...............4........6..........3......8..............72......... 152463897693587241874912653941278536368159724527346189789624315416835972235791468
54 : 1.2.....5..3...........4....4........6..........72.....8......................... 172893465453176892698254173547618329269345718831729546786931254924587631315462987
56 : 1.2.....7..3................4........6...........2.....8.....5.............7..... 192468537653197482478253691845671923269835714317924865784319256921546378536782149
59 : 1.2.....9..3...........6....4........6..........72.....8......................... 172834569653972814498156372945618723267345198831729645386497251729581436514263987
60 : 1.2...5....3................4........6.................8.....7.............81.... 192738564653942187478165293847623915365491728921587346286359471514276839739814652
65 : 1.2..5.....3..............4.4........6.................8.....................6.7. 172495836453681792698732154547218369869374215321569487784953621936127548215846973
66 : 1.2..5.....3..............4.4........6..7...........2..8......................... 172435698493687152856192374347821569269574831518963427984756213725318946631249785
67 : 1.2..5.....3..............6.4..7.....6.................8.......................1. 192765438673824591854391276541673829268159347937482165386217954415936782729548613
68 : 1.2..5.....3..............6.4..7.....6..............1..8......................... 192765438673824591854391276541673829268159347937482615386217954415936782729548163
69 : 1.2..5.....3..............6.4.8......6.................8..7...................... 192465738653287941874913256941856327765132894328749165289574613416328579537691482
71 : 1.2..7.....3..............6.4.7......6.................8..9...................... 152867934693412785874359126348726519269531847517948263485293671926174358731685492
74 : 1.2..9.....3................4..5.....6.............92..8......................... 172539864653842179498671352947258613261397485835164927386425791724916538519783246
81 : 1.29.......3................4........6..............1..8..5.....................8 172934865853672149694185273548213697261597384937846512386459721415728936729361458
82 : 1.29.......3..............8.4.5......6..............1..8......................... 152978463873426951496351278241537896568149732937862514789614325314295687625783149

Leren

<Edit> I've just discovered a bug in my 10 clue PX puzzle searcher and am re-running the job. The number of 10 clue puzzles will be much larger, so the minlexers can hold off if they want until I complete the new run, in a day or so.

Leren

[Mathimagics]

when all 32 transformations are employed, the ED/"PX-different" count drops to 4,208,450

Confirmed.

I rebuilt my PX catalog using these 32 PX-preserving transformations, and I got the same number.

[blue]

when all 32 transformations are employed, the ED/"PX-different" count drops to 4,208,450

Confirmed.

Thank you !

[Leren]

I've been working with Mathimagics who is identifying the ED 10 clue SudokuPX puzzles and solution grids I have found. So far the count is 358 ED grids, 430 ED 10 clue puzzles.

This is work in progress and the counts will be inching up daily. When we have had enough we will make the puzzle list available to others for checking. And no, I haven't found a 9 clue puzzle yet, but I live in hope !

Leren