The New Sudoku Players' Forum

[StrmCkr]

delted

[Pat]

i am pondering if a symetry has been missed from issomphoric identies.

i am wondering if box-box-box relationship was missed. here is an example puzzle i have to demenstrate what i am asking.

Code: Select all

 1 . . | 2 . . | 3 . .
 8 . . | . . . | . 6 1
 . . 7 | . . 8 | . . 9
-------+-------+------
 3 . . | 1 . . | 2 . .
 . 6 . | . . . | . . .
 . . 9 | . . 7 | 4 . 8
-------+-------+------
 . . . | 3 . . | 1 . .
 . 5 . | . . . | . . .
 4 . 8 | . . 9 | . . 7



Boxes 1,5,9 are equal
Boxes 3,4,8 are equal
Boxes 2,6,7 are equal

i have found that you can directly swap any of the equal boxs 1 for 1 and the solution remains unchanged. nothign exciting

but
this property is not listed in the identites of issomorphic puzzles.

the boxes in the answer are equal,
but the boxes in the puzzle differ

[Pat]

Code: Select all

 1 . . | 2 . . | 3 . .
 8 . . | . . . | . 6 1
 . . 7 | . . 8 | . . 9
-------+-------+------
 3 . . | 1 . . | 2 . .
 . 6 . | . . . | . . .
 . . 9 | . . 7 | 4 . 8
-------+-------+------
 . . . | 3 . . | 1 . .
 . 5 . | . . . | . . .
 4 . 8 | . . 9 | . . 7



my next question is this.

if you can rotated combiniation of boxes that are identical and create a puzzle that is identical when considering the affective change. are they equivical.

if you can rotate equal boxs only 90/180/270/360 and or reflect the image of these boxes that is equal and create a new but identical puzzle. { this change seems limited to identical boxes and puzzle state bases}.
is it an issomorphic property?

take this example.

i roated (3,6,9) in box 3 and 4 - 270 degrees cock wise and created a second puzzle.{from the first puzzle} --

Code: Select all

 1 . . | 2 . . | . . 9
 8 . . | . . . | . 6 1
 . . 7 | . . 8 | 3 . .
-------+-------+------
 . . 9 | 1 . . | 2 . .
 . 6 . | . . . | . . .
 3 . . | . . 7 | 4 . 8
-------+-------+------
 . . . | 9 . . | 1 . .
 . 5 . | . . . | . . .
 4 . 8 | . . 3 | . . 7



are they issomorphic given the above condidtions are tracable??

these 2 puzzles are definitely not equivalent

[Pat]

and i'm unable to respond at your other discussion of this

they (SuDoku.Org.UK) have lost my old registration
and don't seem to offer new registrations??

[Mike Barker]

If you submit a solution to one of the daily puzzles, you will be able to reregister. Seems an awkward way to do things, but its the only way I've found that works.

[StrmCkr]

deleted

[Red Ed]

StrmCkr> i am pondering if a symetry has been missed from issomphoric identies.

You're thinking of the set, let's call it S, of 9! x 3359232 validity-preserving transformations. That set consists of relabelling (9!) and cell-moving (3359232) operations that transform any valid sudoku solution into another valid sudoku solution.

Your box-box operations only preserve the validity of certain special solution grids (like the MC grid); they are not validity-preserving for all solution grids. That's why they're not listed in S.

[StrmCkr]

[deleted]

[Pat]

i have another question that starts with a documents i have read on this subject..

that some roations/reflections arent valid on some grids.

but row/column swaping/ number changing works on all.

i have grids that cant rotate or reflect:
but you can permate the numbers/columns/rows leaves valid grids as well.

so how would it be a properity of all grids if rotation or relfections dont work on some grids?

rotation and reflection do preserve the equivalence

[Red Ed]

StrmCkr, please show us a grid that you think can't be rotated or can't be reflected. I expect that by explaining your mistake on a particular example, we can help you understand some of the general issues too.

My money's on this being the old isomorphism/automorphism point of confusion again.

[coloin]

I dont think your example is unusual - all you have done is moved a few clues around - and perhaps coincidently the puzzle is valid. The two puzzles are not equivalent.

Code: Select all

+---+---+---+
|1..|2..|..9|
|8..|...|.61|
|..7|..8|3..|
+---+---+---+
|..9|1..|2..|
|.6.|...|...|
|3..|..7|4.8|
+---+---+---+
|...|9..|1..|
|.5.|...|...|
|4.8|..3|..7|
+---+---+---+ original [non minimal] puzzle

heres another puzzle with clues in B1 and B5 moved around.

Code: Select all

+---+---+---+
|7..|2..|..9|
|8..|...|.61|
|..1|..8|3..|
+---+---+---+
|..9|..1|2..|
|.6.|7..|...|
|3..|...|4.8|
+---+---+---+
|...|9..|1..|
|.5.|...|...|
|4.8|..3|..7|
+---+---+---+

Ive noticed that most puzzles [your example included] have many other puzzles "near" in a {-1+1} operation.

Some puzzles are remote - a good example of this is the newly found 17 puzzles...... except when I searched hard I did find this 21-puzzle.

Code: Select all

+---+---+---+
|...|...|...|
|12.|3..|8..|
|34.|12.|...|
+---+---+---+
|...|.6.|..7|
|..9|..7|..8|
|6..|...|5..|
+---+---+---+
|..5|..3|..4|
|.1.|...|...|
|...|8..|.9.|
+---+---+---+ untouchable-21

There are no other puzzles "near"

Conversly your puzzle [with the only 2 superfluos clues at r1c9 and r2c1 removed]

Code: Select all

+---+---+---+
|1..|2..|...|
|...|...|.61|
|..7|..8|3..|
+---+---+---+
|..9|1..|2..|
|.6.|...|...|
|3..|..7|4.8|
+---+---+---+
|...|9..|1..|
|.5.|...|...|
|4.8|..3|..7|
+---+---+---+

All clues apart from 4@r1c9 can be replaced with another clue somewhere to give a total of 137 other valid puzzles within {-1+1}.

C

[StrmCkr]

removed

[coloin]

Ah, I see, B3,B4 and B8 are the same.

Code: Select all

+---+---+---+
|195|276|384|
|842|953|761|
|637|418|529|
+---+---+---+
|384|195|276|
|761|842|953|
|529|637|418|
+---+---+---+
|276|384|195|
|953|761|842|
|418|529|637|
+---+---+---+

In fact all 6 bands are equivalent.

This is an automorphic grid.

[StrmCkr]

deleted

[coloin]

Here is a reference to a list of the automorphic grids.

I still think you are changing clues in boxes.

Why have you chosen to rotate 2 boxes and reflect the other ?

the same puzzle correct? all having the same solution.

- no probably many different solutions.

C