The New Sudoku Players' Forum

[Leren]

Code: Select all

*-----------*
|..1|..9|6.4|
|36.|..1|...|
|...|65.|3..|
|---+---+---|
|6.7|.2.|...|
|8.9|...|...|
|.1.|...|9.8|
|---+---+---|
|1..|8..|5.7|
|.73|1..|...|
|...|.47|..3|
*-----------*
..1..96.436...1......65.3..6.7.2....8.9.......1....9.81..8..5.7.731.........47..3

[eleven]

The puzzle has a digit symmetry, which is rarely known, though it is the most common under the (automorph) solution grids.

(1)(2)(3)(4,5)(6,7)(8,9) are the symmetric digits.
Swap bands 1,3, then in each stack the first and 3rd column

Code: Select all

*-----------*      |---+---+---|    |---+---+---|
|..1|..9|6.4|      |1..|8..|5.7|    |..1|..8|7.5|
|36.|..1|...|      |.73|1..|...|    |37.|..1|...|
|...|65.|3..|      |...|.47|..3|    |...|74.|3..|
|---+---+---|      |---+---+---|    |---+---+---|
|6.7|.2.|...|  ->  |6.7|.2.|...| -> |7.6|.2.|...|
|8.9|...|...|      |8.9|...|...|    |9.8|...|...|
|.1.|...|9.8|      |.1.|...|9.8|    |.1.|...|8.9|
|---+---+---|      *-----------*    *-----------*
|1..|8..|5.7|      |..1|..9|6.4|    |1..|9..|4.6|
|.73|1..|...|      |36.|..1|...|    |.63|1..|...|
|...|.47|..3|      |...|65.|3..|    |...|.56|..3|
*-----------*      *-----------*    *-----------*


If you do the same with the solution of the puzzle (swap bands and columns, and change the digits), you have a solution for the same puzzle again.
If it is not the same, the puzzle has 2 solutions. So for a unique puzzle the solution must have the same symmetry too.

You can immediately place 7 digits (in the 9 cells, which do not change by the transformation), and solve with singles.

[Cenoman]

I suspected the sticks symmetry, that I discovered in 2018 (here) but was unable to bring it to light... Leren's and eleven's teaching have been forgotten.
So thank you for this wake-up call !

I just wonder how it is humanly possible to find the sticks symmetry when it has been mingled by an isomorphism, assigning not-centered invariant cells ?

e.g. which approach to use, if Leren had posted this puzzle ?
..7.8..4.89..2.7..41.7...2......897....17...45....3..6..18.23.............96..4.2

Code: Select all

*---+---+---*
|..7|.8.|.4.|
|89.|.2.|7..|
|41.|7..|.2.|
*---+---+---*
|...|..8|97.|
|...|17.|..4|
|5..|..3|..6|
*---+---+---*
|..1|8.2|3..|
|...|...|...|
|..9|6..|4.2|
*---+---+---*

[Leren]

Code: Select all

*-------------------------------------------------------------*
|*2-36  *25-36   *7       | 359  8     1569 | 156  4     1359 |
| 8      9        356     | 345  2     1456 | 7    1356  135  |
| 4      1        356     | 7    3569  569  | 568  2     3589 |
|-------------------------+-----------------+-----------------|
| 1      2346     2346    | 245  456   8    | 9    7     35   |
| 9      2368     2368    | 1    7     56   | 258  358   4    |
|*5     *7       *2-48    | 249  49    3    | 128  18    6    |
|-------------------------+-----------------+-----------------|
| 67     456      1       | 8    459   2    | 3    569   57   |
|*27-36 *25-3468 *25-3468 | 3459 13459 4579 | 1568 15689 1578 |
| 37     358      9       | 6    135   57   | 4    158   2    |
*-------------------------------------------------------------*

Row Sticks Symmetry : [19] [36] [48] [2] [5] [7] . Elimination Rows 168 Columns 1-3 => r158c123 = 257; stte

PS It should be obvious that you can't hide Stick Symmetry by morphing a puzzle, because the three elimination mini-row/columns always end up in the one chute in the three boxes of that chute.

Leren

[Cenoman]

PS It should be obvious that you can't hide Stick Symmetry by morphing a puzzle, because the three elimination mini-row/columns always end up in the one chute in the three boxes of that chute.

Of course, Leren. But my point was not what is the solution ? but how does one come manually to a solution ?, notably, how can one bring to light an automorphism proving the sticks symmetry ? How did you identified the three stable mini-rows r168c123, and the digit symmetry (1)(2)(3)(4,5)(6,7)(8,9) ?
I am much more interested in the mental process, than in the results. In one word, what is the algorithm ?

[eleven]

Since digit symmetric puzzles are very rare, normally it is not worth to look for them. But if you know, that it is such a puzzle, you can find the symmetry also, when the puzzle is mangled (not in "normal form"), some times easier, some times harder.
You have to look at the box patterns to check, which boxes can be symmetric to one another. In your sample, these are only b23, b67 and b89, so it is easy. The only symmetry, which is possible, is the sticks symmetry then.
Then look, which digits can be symmetric. 8 would have to go to 4, 9 to 1, 7 fixed. Then 2 fixed. Then 3 to 6, and 5 fixed. All fits.

[Leren]

Cenoman wrote : In one word, what is the algorithm ?

Hi Cenoman, nice oxymoron there, that's four words, not one. I'm a great fan of oxymorons, or perhaps I should say I'm an extremely average fan of them

Seriously though, I would never have understood Sticks Symmetry had it not been for this wonderful post by Mauricio, many years ago. I've kept it in my solver, and I can't seem to find it on the forum now , it may have been removed.

Code: Select all

' Sticks symmetry - example by Mauricio

'9 . .|. . 3|. . 6    b a c|. . .|. . .
'. 6 .|. . 1|. 5 .    b a c|. . .|. . .
'. . 4|6 . .|2 . .    b a c|. . .|. . .
'-----+-----+-----    -----+-----+-----
'. 7 .|. 1 .|9 . 8    d X d|. X .|. X .
'. . .|. . .|. . .    d X d|. X .|. X .
'3 . 2|. . .|. 7 .    d X d|. X .|. X .
'-----+-----+-----    -----+-----+-----
'. . 8|2 . .|5 . .    c a b|. . .|. . .
'. 5 .|1 . .|. 6 .    c a b|. . .|. . .
'4 . .|. . 5|. . 3    c a b|. . .|. . .
'
'BxCx (exchange bands 13 and Columns 13,46,79) + Isomorphism '(1)(4)(7)(23)(56)(89)

So, in seven words : This is an example of the algorithm.

If you carry out the column and band swaps like Mauricio says, the paired digits are consistently mapped to their partner digit, and the unpaired digits are mapped to themselves, or not changed.

If there are 3 sets of paired digits, and 3 unpaired digits, the 3 unpaired digits are confined to the 3 unswapped columns in the unswapped band (indicated by the X's in the exemplar diagram on the right).

So, this is what I get for this example :

Code: Select all

*---------------------------------------------------------*
| 9     128 157  | 4578    24578 3      | 1478  148  6    |
| 278   6   37   | 4789    24789 1      | 3478  5    479  |
| 1578  138 4    | 6       5789  789    | 2     1389 179  |
|----------------+----------------------+-----------------|
| 56   *7   56   | 34     *1     24     | 9    *4    8    |
| 18   *14  19   | 345789 *47    246789 | 1346 *14   1245 |
| 3    *14  2    | 4589   *4     4689   | 146  *7    145  |
|----------------+----------------------+-----------------|
| 167   139 8    | 2       34679 4679   | 5     149  1479 |
| 27    5   379  | 1       34789 4789   | 478   6    2479 |
| 4     129 1679 | 789     6789  5      | 178   1289 3    |
*---------------------------------------------------------*

Column Sticks Symmetry : [23] [56] [89]   [1] [4] [7] . Elimination Rows 4-6 Columns 258.

Once you get the hang of this example, generalising it for other directions isn't too hard. Leren

[eleven]

Seriously though, I would never have understood Sticks Symmetry had it not been for this wonderful post by Mauricio, many years ago. I've kept it in my solver, and I can't seem to find it on the forum now , it may have been removed.

The puzzle was posted here, and i have changed it to that presentation here.

[Leren]

Hi eleven, it seems then that I have you to thank for that post. For me the best way to understand something new is a worked example.

Leren

[StrmCkr]

The puzzle has a digit symmetry, which is rarely known, though it is the most common under the (automorph) solution grids.

(1)(2)(3)(4,5)(6,7)(8,9) are the symmetric digits.
Swap bands 1,3, then in each stack the first and 3rd column

Code: Select all

*-----------*      |---+---+---|    |---+---+---|
|..1|..9|6.4|      |1..|8..|5.7|    |..1|..8|7.5|
|36.|..1|...|      |.73|1..|...|    |37.|..1|...|
|...|65.|3..|      |...|.47|..3|    |...|74.|3..|
|---+---+---|      |---+---+---|    |---+---+---|
|6.7|.2.|...|  ->  |6.7|.2.|...| -> |7.6|.2.|...|
|8.9|...|...|      |8.9|...|...|    |9.8|...|...|
|.1.|...|9.8|      |.1.|...|9.8|    |.1.|...|8.9|
|---+---+---|      *-----------*    *-----------*
|1..|8..|5.7|      |..1|..9|6.4|    |1..|9..|4.6|
|.73|1..|...|      |36.|..1|...|    |.63|1..|...|
|...|.47|..3|      |...|65.|3..|    |...|.56|..3|
*-----------*      *-----------*    *-----------*


If you do the same with the solution of the puzzle (swap bands and columns, and change the digits), you have a solution for the same puzzle again.
If it is not the same, the puzzle has 2 solutions. So for a unique puzzle the solution must have the same symmetry too.

You can immediately place 7 digits (in the 9 cells, which do not change by the transformation), and solve with singles.

digits 123 are a fixed tuple set as these digits are the only ones that didn't undergo any digit changes from first and last transformation
{ discovered when we put the grid back to starting position by the last step} .

we know that the center band specifically Mini Col 2 in each of the boxs {4,5,6} did not change during any of the transformations thus these cells also hold the set that also did not change. {9 cells - 3 sets of 3 digits}

From that you can deduce that R5C5 = 1, and R6C5 = 3 , then use these digits to solve R4C8 = 1, R5C8=3, R6C8 = 2, R4C2 = 3, R5C2=2 R1C1 = 2, R1c3 = 3 , r7c6 =3,r9c7=1, r3c9=1,r7c3=2,r3c6=2,r9c4=2

Code: Select all

.---------------.--------------.----------------.
| 2    58   1   | 3    78  9   | 6    578   4   |
| 3    6    458 | 47   78  1   | 278  5789  259 |
| 479  489  48  | 6    5   2   | 3    789   1   |
:---------------+--------------+----------------:
| 6    3    7   | 459  2   458 | 4    1     5   |
| 8    2    9   | 457  1   456 | 47   3     56  |
| 45   1    45  | 457  3   456 | 9    2     8   |
:---------------+--------------+----------------:
| 1    49   2   | 8    69  3   | 5    469   7   |
| 459  7    3   | 1    69  56  | 248  4689  269 |
| 59   589  568 | 2    4   7   | 1    689   3   |
'---------------'--------------'----------------'

'

singles to the end.

[Cenoman]

Many thanks to you, Leren and eleven, for your teaching efforts

This is an example of the algorithm.

Mauricio's puzzle didn't help me more than yours, since it has the same "normal form" (thus the same automorphism)

But eleven gave the tricks:

look at the box patterns to check, which boxes can be symmetric to one another. In your sample, these are only b23, b67 and b89
...and...
then look, which digits can be symmetric

This made me aware that in the sticks symmetry, there must be a symmetry between six box patterns (in two bands or two stacks).
With your indications and with the link to the thread "about-red-ed-s-sudoku-symmetry-group", the ball is in my court now !

[Cenoman]

This post is definitely off-topic, as regards sudoku puzzles (whether symmetric or not)

nice oxymoron there, that's four words, not one. I'm a great fan of oxymorons, or perhaps I should say I'm an extremely average fan of them.

Hi Leren,
Your litterary comment made me read again my book "Figures of speech" ("Les figures de style" in French)
In French, with have the same name "oxymore" for a figure of speech, linking words with opposite meanings, e.g. 'extremely average'. 'In one word', when you actually use several, would rather be a "litote", that my translator says to be an 'understatement'. Note that the same English word 'understatement' is used to translate the French "euphémisme". Both are weakening figures, with a slight difference: the first one suggests more than actually said, the second tries to hide something unpleasant.

The French expression "En un mot", very currently used, means that one is going to summarise. It would be better translated in English to: 'in summary', or 'in short', or 'in a word', or 'in a nutshell' (I am an extremely average fan of metaphors !)

Sorry for this pedandic post. I'm just killing time !
PS: never worked as a language teacher.

[StrmCkr]

Thought it translated to, "said in a moment" ie quickly said, But its been decades since I've practiced french.

Anyone have a quick link to the symmetrical solving techniques so i might look at adding it to my solver. I had it and cant find it today forgot to bookmark

http://forum.enjoysudoku.com/about-red-ed-s-sudoku-symmetry-group-t6526.html

Found it