The New Sudoku Players' Forum

[swb01]

The minlex of the puzzles are uniform for all boxes: MinLexPuzzle - 6, Puzzle1 - 3, Puzzle2 - 3.
The produced grids follow the pattern shown below for Boxes 1-3.

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Box1 - (Grid1 produced six times):
Cell     Options  Minimal?   Grid         Puzzle
  1        5        N        Grid1        Puzzle1
  1        6        Y        MinMaxGrid   MinMaxPuzzle
  1        9        Y        Grid2        MinMaxPuzzle
  2        6        N        Grid1        Puzzle2
  4        9        N        Grid1        Puzzle2
  6        3        Y        Grid2        MinMaxPuzzle
  6        8        N        Grid1        Puzzle1
  6        9        Y        MinMaxGrid   MinMaxPuzzle
  8        2        N        Grid1        Puzzle1
  8        3        Y        MinMaxGrid   MinMaxPuzzle
  8        6        Y        Grid2        MinMaxPuzzle
  9        3        N        Grid1        Puzzle2
Box2 - (Grid2 produced six times):
Cell     Options  Minimal?   Grid         Puzzle
  1        4        Y        Grid1        MinMaxPuzzle
  1        6        N        Grid2        Puzzle1
  1        7        Y        MinMaxGrid   MinMaxPuzzle
  2        4        N        Grid2        Puzzle2
  4        7        N        Grid2        Puzzle2
  6        1        Y        MinMaxGrid   MinMaxPuzzle
  6        7        Y        Grid1        MinMaxPuzzle
  6        9        N        Grid2        Puzzle1
  8        1        Y        Grid1        MinMaxPuzzle
  8        3        N        Grid2        Puzzle1
  8        4        Y        MinMaxGrid   MinMaxPuzzle
  9        1        N        Grid2        Puzzle2
Box3 - (MinMaxGrid produced six times):
Cell     Options  Minimal?   Grid         Puzzle
  1        4        N        MinMaxGrid   Puzzle1
  1        5        Y        Grid2        MinMaxPuzzle
  1        8        Y        Grid1        MinMaxPuzzle
  2        5        N        MinMaxGrid   Puzzle2
  4        8        N        MinMaxGrid   Puzzle2
  6        2        Y        Grid1        MinMaxPuzzle
  6        7        N        MinMaxGrid   Puzzle1
  6        8        Y        Grid2        MinMaxPuzzle
  8        1        N        MinMaxGrid   Puzzle1
  8        2        Y        Grid2        MinMaxPuzzle
  8        5        Y        Grid1        MinMaxPuzzle
  9        2        N        MinMaxGrid   Puzzle2


[coloin]

Most of that i agree with ..... apart from column 4 !
By definition, a puzzle always maps to a grid solution.

Here, If the puzzle is minimal it is an isomorph of the MinMaxPuzzle and the grid solution is an isomorph of that MinMaxPuzzleGrid
There are 56 isomorphs of the puzzle mapping to the 56 isomorphs of the grid solution... hence the solution grid must have at least 56 automorphic varations.

[swb01]

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MinMaxPuzzleGrid: 123456789457189236968372514291738465374265198685941327546813972732694851819527643 ... MinLex of the grid produced by the MaxMinLex puzzle.
MinMaxGrid:       681792453249351867735846912912465738453987621867213549576129384198534276324678195 ... Grid produced by the MaxMinLex puzzle.               
Grid1:            561492873948357162723816954432561798657948321819723546276135489195684237384279615               
Grid2:            981642573243759168765831924132468759654927831879513642426195387597384216318276495 

coloin wrote: Here, If the puzzle is minimal it is an isomorph of the MinMaxPuzzle ...

Yes, for each box, six of the puzzles made by adding one of the optional values are minimal and isomorph to the MinMaxPuzzle, the non-minimal puzzles, once minimized, isomorph to Puzzle1 and Puzzle2, three each.

and the grid solution is an isomorph of that MinMaxPuzzleGrid.

Yes. The grid solutions for the minimal puzzles are isomorphic to the MinMaxPuzzleGrid. (The grid solutions of the non-minimal solutions are also isomorphic to the MinMaxPuzzleGrid.)

There are 54 isomorphs of the puzzle mapping to the 54 isomorphs of the grid solution... (typo corrected)

Only three grid solutions are produced, labeled MinMaxGrid, Grid1 and Grid2 above. The 54 isomorphs of the puzzle produce 3 grids, all isomorphs of the MinMaxPuzzleGrid.

hence the solution grid must have at least 54 automorphic variations.

In my March, 7, 2021 comment I show 36 automorphic variations for the MinMaxPuzzleGrid (and its isomorphs as expected). One for the MinMaxPuzzleGrid is the identity transformation. Perhaps there are 18 more in which case, my grid minlexer didn't find them all.

[coloin]

Well certainly confusing !!!
However ... the confusion is apparent as all the 108 puzzles have essentially the same solution grid ... well done for spotting that !!!

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5.1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5         
.61..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5         
..14.2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5 minimal 
561492873948357162723816954432561798657948321819723546276135489195684237384279615                                                                                           
                                                                                           
6.1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5 minimal 
..1..24.3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5         
..1..2.53.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5         
681792453249351867735846912912465738453987621867213549576129384198534276324678195

..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7.1.5  largest known min lex minimal puzzle !!!
                                                                                             
                                                                                           
9.1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5 minimal
..16.2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5         
..1.42..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7...5         
981642573243759168765831924132468759654927831879513642426195387597384216318276495


all essentially the same

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561492873948357162723816954432561798657948321819723546276135489195684237384279615
681792453249351867735846912912465738453987621867213549576129384198534276324678195
981642573243759168765831924132468759654927831879513642426195387597384216318276495


I suppose my statement should be that this grid solution has at least 18 automorphs ...

[swb01]

I agree, the maxminlex puzzle is interesting. ... I'm still trying to divine the theorem being applied here.

coloin wrote on May 5:
There are three solution grids to the 27 clue sub-puzzle. Each box is essentially the same.

Is there a definition of essentially the same boxes?
(For this 27 clue sub-puzzle, each box, row and column have 15 options of pattern 222333.)
I also noticed for the sub-puzzle that no reductions can be made to the 135 options based on logical examination (I don't know the correct term for "logical examination" versus trial-and-error or guessing.) However, if any one options is added, the solution can be made with only logical examination - every cell can be determined.

Now for the theorem, is this close?
For a SuDoKu sub-puzzle with essentially the same boxes, if by adding one option there are x ways to make the same isomorphic puzzle, then the solution grid has at least x automorphisms.

[coloin]

your theorm was what I thought ... except there were 54 ways to make the isomorphic puzzle - and you found 36 automorphisms in the solution grid !!
There were 18 ways to make the "same" puzzle solution - so i suggested there might be at least 18 automorphisms ... must be true !!

maybe you will find information and links in About Red Ed's Sudoku symmetry group Page 8 gives the automorphic grids table

Meanwhile I have found a few puzzles to challange the max min lex [minimal] puzzle ... but the minlexing software lets me down I think !!!!!

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..1..2..3.4..5..6.7..8..9....2..57...5.9....18...13.4...61...8..9...42..3...7...5
..1..2..3.4..5..6.7..8..9....2..57...5.9....18...13.4...61..3...9...4..63...7.19.
..1..2..3.4..5..6.7..8..9....2..57...5.9....18...13.4...61..3...9...4.763...7..9.
..1..2..3.4..5..6.7..8..9....2..57...5.9....18...13.4...61..3.4.9...4..63...7..9.
..1..2..3.4..5..6.7..8..9....2..57...5.9.7..18...1..4...61...8..9...42..3...7...5
..1..2..3.4..5..6.7..8..9....2..57...5.9.7..18...1..4...61..3...9...4..63...7.19.
..1..2..3.4..5..6.7..8..9....2..57...5.9.7..18...1..4...61..3...9...4..63...7.19.
..1..2..3.4..5..6.7..8..9....2..57...5.9.7..18...1..4...61..3...9...4.763...7..9.
..1..2..3.4..5..6.7..8..9....2..57...5.9.7..18...1..4...61..3.4.9...4..63...7..9.
..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7.1.5  and this is the original maxminlex puzzle
above all same solution grid !!!

..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.46..4.7...5.7.1...8.1.5..42..
..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.46..4.7...51....42..27.1...8.
..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.46.7.1...8..84.7...51....42..



[swb01]

Thank you for the reference for information on automorphic grids.

On your three cases that challenge the max min lex [minimal] puzzle:
..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.4...61...8..9...42..3...7.1.5 - MaxMinLex
..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.46..4.7...5.7.1...8.1.5..42.. #1
..1..2..3.2..4..5.4..13.6....32..7...5...4..86...1..9...7..5.1..8..6.2..9..3...74 #1 minlex, Transposition = Yes, Digits ==> 469137258, Rows ==> 213546879, Columns ==> 132798465
..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.46..4.7...51....42..27.1...8. #2
..1..2..3.2..4..5.4..13.6....32..7...5...4..86...1..9...7..581..8..6.2..9..3....4 #2 minlex, Transposition = Yes, Digits ==> 137258469, Rows ==> 321654987, Columns ==> 321897654
..1..2..3.4..5..6.7..8..9....2.6.7...5.9....18....3.46.7.1...8..84.7...51....42.. #3
..1..2..3.2..4..5.4..13.6....32..7...5...4..86...1..9...7..5.1..8..6.2.99..3....4 #3 minlex, Transposition = Yes, Digits ==> 258469137, Rows ==> 132465798, Columns ==> 213789546