## is a symertry missing from issomorphic variations

Everything about Sudoku that doesn't fit in one of the other sections
Indeed !

I think what strmckr is doing is going from this puzzle

Code: Select all
`+---+---+---+|1..|2..|3..||8..|...|.61||..7|..8|..9|+---+---+---+|3..|1..|2..||.6.|...|...||..9|..7|4.8|+---+---+---+|...|3..|1..||.5.|...|...||4.8|..9|..7|+---+---+---+`

[which coincidently happens to have a solution grid with 6 equivalent bands]

to this [different] valid puzzle
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|+---+---+---+`

Now it seems that a few clues have been swapped around.......but I believe strmckr is able to follow these swaps in B3 and B4 and then is able to determine what the swap should be in B8 which defines a new valid puzzle.

Hows that done then ?

C
coloin

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Joined: 05 May 2005

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Last edited by StrmCkr on Sat Dec 13, 2014 6:06 am, edited 1 time in total.
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StrmCkr

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### re: equivalence

StrmCkr wrote:i'm showing simplistic moves off how is see the manipulation compared to that of issomorphic properties and pointing out how they are one and the same.

i will go more in detail when im completed the simplistic version of the 2x2 puzzle i was listing as examples to show the diffrence in each puzzle and how they are all related to the concesus view of issomorphic properties by example.

i belive the same things happen in larger scale puzzles (9x9) with the exception that the degree of complexity of known positions alters or directly limits the avialablity of direct: box-box rotation which leaves the similar locked" positions but altered by a row/column simplly by rotation of box a= "zero solutins" by rotations of line of sight candidates = 1 solution again.

where as alll the things im suggesting is applicable.

i will hopefully shed some light on how it is accomplished.
which also should show that i am correctly perposing something new to issomorphs not seen befor.

im purposing that symetry is missing

or that the ones generally used to identify issomphs is perhaps only a partail view of what transcribes an issomprh from each other.

i belive the general view is more along the lines a limited view of transformation of grid a- b.

by manipulating the rows/columns in question as a single entity rather than the fact a row/column is formed by a combination of line of sight from boxs 3 boxes.

i belive the full view is actuall. combination of the options. like i'm showing in the simplistic grids on page 2.

where they are actually options affects (like i have shown) between boxs.

i have shown that the puzzles post above by coloin are identical to each other by 2 diagonal axis rotations in two specific boxes (90 degrees) box 3&4

where the third box roates 180 degrees (reflects it self) box 8
= the moves can be reversed and the puzzles reverts back to the original.

the key point is that no extra clue is added or removed.

hi StrmCkr

you are starting from one puzzle and somehow creating another puzzle
the transformation you're using
does not preserve the equivalence
the new puzzle is not guaranteed to be solvable by the same logic used in solving the original puzzle

Pat

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Joined: 18 July 2005

StrmCkr wrote:im purposing that symetry is missing

Wrong terminology! Please take note (again):
A "symmetry" is described by two things together ...
• A permutation of the digits 1 to 9. This covers digit relabelling.
• A permutation of the values 1 to 81. This covers cell swapping.
... with the proviso that it does not affect the validity of any grid.

The operations you are talking about generally do affect the validity of some grids (making valid ones invalid or, sometimes, invalid ones valid). So they are not symmetries, they're just localised puzzle-specific cell swaps.

How your investigations might usefully get back on course:
As far as I can see, all you've found is that some pairs of puzzles are linked not just by arbitrary clue addition/removal (as is usually the case), but by some rather nice-looking localised cell swaps. So why don't you try to work out if there's anything special about the corresponding solution grids? We might conjecture that, more often than expected at random, the grids are automorphic. I guess it would be nice to know the answer to that, though it would be far quicker to check using a specialist computer program than (as I guess you are doing) with an ordinary solver.
Red Ed

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StrmCkr

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Can you see that this process works in every puzzle ?
Code: Select all
`+---+---+---+|5..|...|..9||.2.|1..|.7.||..8|...|3..|+---+---+---+|.4.|..2|...||...|.5.|...||...|7.6|.1.|+---+---+---+|..3|...|8..||.6.|..4|.2.||9..|...|..5|+---+---+---+ puzz1`

This puzzle is highly symetrical - the 3&9 and the 1&4 are 5&8 isomorphically interchangeable !

The 7 and the 6 fulfil very similar roles too

Puzz1>Puzz2 different puzz-a
Puzz1>puzz3 different puzz-b
Puzz1>puzz4 swaps c7c9 plus r7r9 [isomorph]
puzz4>puzz5 is relabelling 3&9 [isomorph]
puzz5>puzz6 is relabelling 5&8 [isomorph]

The operations will always be reversable

I think you need to demonstrate the effects of the puzz1>2,and puzz1>3. swaps in a less symetrical puzzle.

I can see that a clue which is swapped will still have the same line interactions in one of the directions. What I cant see is how you know that a few swaps will complete the puzzle !

C
coloin

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StrmCkr wrote:i found a way to work this for every puzzle.

Since you haven't bothered to define "this" with any precision, I don't know how we're supposed to take you seriously.

You appear to be making the claim that:
• For any puzzle, it is possible to do <something you haven't bothered to define> to it to create a new, valid, non-isomorphic puzzle.
Big deal.
Red Ed

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StrmCkr

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Red Ed wrote:
StrmCkr wrote:i found a way to work this for every puzzle.

Since you haven't bothered to define "this" with any precision, I don't know how we're supposed to take you seriously.

You appear to be making the claim that:
• For any puzzle, it is possible to do <something you haven't bothered to define> to it to create a new, valid, non-isomorphic puzzle.
Big deal.

I'm guessing the "this" is using "same solution up to labeling" for equivalence

StrmCkr, although it can sometimes be visually appealing, using properties of an object
as an equivalence measure will almost always fail to meet the requirements of mathematical equivalence

properties can always be used to prove inequivalence
but finding properties (like "solution path" for sudoku) that prove equivalence is a tall order

one property of sudoku puzzles that does meet the equivalence requirements is the minlex grid representation
gsf
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StrmCkr

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StrmCkr wrote:
one property of sudoku puzzles that does meet the equivalence requirements is the minlex grid representation

can you explain what you are suggesting a little better glenn?

i can read that a few ways. as a suggestion to take the minlex grids and explain it off that grid so one grid is related to another and vice versa.

or something else.

equivalence is exact -- either two items are the same or they are not
equivalence often does not help with "similarity"
this is readily shown by comparing minlex grids of slightly altered puzzles, like the two you proposed above
Code: Select all
`000000000001002003040050060000000700002003540050068001003040002008001350900700000000000000001002003040050060000000700002003501050068040003001002008040350900700000`

you are trying to analyze how puzzles are similar by measuring some property
(in your case you compare solution paths)
two puzzles having the same solution path is insufficient to prove equivalence

in mathematics symmetry, isomorphism and equivalence are tightly related
this is why you are getting some flack trying to explain a new "symmetry"

so call it something else besides symmetry (e.g. similarity)
and you'll be clear of conflicts with mathematic equivalence/isomorphism/symmetry
gsf
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It's not so much the mistaken use of "isomorphism" etc. that cheeses me off, it's the random splattering of terms like that into StrmCkr's postings (viz. "a postulation that clues are not fixed into a single solution at all. but instead many options of solutions each being unique but not all equal to one anoter visible or via current state of issomorphic properties.") that make me despair of ever understanding what he's proposing. He's probably observed an interesting feature; I just can't discern what it is or therefore how to evaluate its significance.
Red Ed

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StrmCkr

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