The New Sudoku Players' Forum

[shye]

hello, i have an interesting puzzle that i'd like to check my logic is valid for

Code: Select all

+-------+-------+-------+
| . . 5 | . 3 . | . . . |
| 4 . . | 1 . . | . . . |
| . . . | . . 6 | . 9 8 |
+-------+-------+-------+
| . . . | . 7 . | 5 . . |
| . 9 3 | . . . | . . 2 |
| 8 . 7 | . 5 . | 3 1 . |
+-------+-------+-------+
| . . . | . . 8 | . 6 . |
| . 7 . | . . . | 2 . . |
| 9 . . | . 6 5 | . . 1 |
+-------+-------+-------+
..5.3....4..1..........6.98....7.5...93.....28.7.5.31......8.6..7....2..9...65..1

estimated rating: 7.2

before starting the puzzle, if we were to take every cell in this grid and think of it as rXcY=Z then transform it into rXcZ=Y, the resulting puzzle will look identical. that is to say, every given in this puzzle indexes to a given digit in a looping pattern
eg:
r1c3=5 => r1c5=3 => r1c3=5 loop
r6c5=5 => r6c5=5 loop

so much like other forms of GSP, if this puzzle has a unique solution (which it does) that solution must be comprised of indexing cycles (which it does)

Code: Select all

.------------------.-------------------.-----------------.
| 167   16+8  5    | 489     3    479  | 1467 *2     467 |
| 4     268  *9    | 1       28   27   | 67    35    5+3 |
| 1237  123   12   |*5       2+4  6    | 147   9     8   |
:------------------+-------------------+-----------------:
| 126   1246  1246 | 234689  7    2349 | 5     48    469 |
|*5     9     3    | 468     48+1 14   | 4678  478   2   |
| 8     246   7    | 2469    5    249  | 3     1     469 |
:------------------+-------------------+-----------------:
| 123  *5     124  | 2347    14+2 8    |*9     6     34+7|
| 136   7     1468 | 34     *9    134  | 2     3458  34+5|
| 9     234   248  | 2347    6    5    | 478   3478  1   |
'------------------'-------------------'-----------------'

all the digits gained at the start (marked with *s) therefore reveal enough singles to finish the puzzle

but although i cant see a fault in my logic here, i'm unsure still if this is completely sound. it might be that i just made a special case puzzle, so if anyone could provide a counterexample, or verify this logic holds, or just clear up any loose ends that would be much appreciated!

also if this is valid, is this form of symmetry already known and i just haven't heard of it before?

[StrmCkr]

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

are the known types
I'll toss this in my solver and see if it hits on anything but others are better at this kind of stuff then me.

Not sure what the first equation is doing

Rxcy = z

So your swapping digit id (z)
For col location (Y)
?? Confused on that..

think of a standard grid as 3 arrays:

Map out the grid as 81 cells 9x9

[1-9] as row storing 9 specific cells from the map
[1-9] as cols storing 9 specific cells from the map

[1-9] digit storing cells from digits of a loaded grid.

When you change the grid issomorphically :

index number[1..9] of R changes with index number [1..9] of R
or
index number [1..9] of C changes with index number [1..9] of C

- the cells assigned of the index number don't change..


when you change digits
index [1..9] changes with index [1..9] but the specific cells don't.

The resulting change when looking at r,c the r*c index shows what cell is now in rc due to the changes applied.

So I'm a bit stumped on what you are doing.
if i swap RxCy = Z for RxCz = y

Code: Select all

.-----------------------.------------------------.---------------------.
| 246789  46789  3      | 12789   5       12789  | 1246   12467  12467 |
| 1       56789  256789 | 4       23789   23789  | 2356   23567  23567 |
| 2457    457    2457   | 1237    1237    6      | 12345  8      9     |
:-----------------------+------------------------+---------------------:
| 4689    4689   4689   | 123689  5       123489 | 7      12346  12346 |
| 45678   2      3      | 1678    14678   1478   | 1456   1456   9     |
| 1       469    3      | 269     5       249    | 7      8      246   |
:-----------------------+------------------------+---------------------:
| 34579   34579  4579   | 12379   123479  6      | 12345  8      12345 |
| 345689  2      45689  | 1389    13489   13489  | 7      13456  13456 |
| 1       3478   478    | 2378    5       6      | 234    234    9     |
'-----------------------'------------------------'---------------------'

thats the error grid i get...

[shye]

thanks for taking the time to look at this

im unsure how you got that grid, for me it looks like so:

Code: Select all

 rXcY=Z                                                              rXcZ=Y                        (boxes do not operate the same here)
.-------------------.---------------------.-------------------.     .---------------------.---------------------.---------------------.
| 1267  1268   5    | 24789   3     2479  | 1467  247   467   |     | 127   12468  5      | 46789   3    1279   | 146789  24    46    |
| 4     2368   2689 | 1       289   279   | 67    2357  3567  |     | 4     23568  289    | 1       89   2379   | 6789    235   356   |
| 1237  123    12   | 2457    24    6     | 147   9     8     |     | 1237  12345  12     | 457     4    6      | 147     9     8     |
:-------------------+---------------------+-------------------:     :---------------------+---------------------+---------------------:
| 126   1246   1246 | 234689  7     12349 | 5     48    469   |     | 1236  12346  46     | 234689  7    12349  | 5       48    469   |
| 156   9      3    | 468     148   14    | 4678  478   2     |     | 156   9      3      | 45678   1    147    | 78      4578  2     |
| 8     246    7    | 2469    5     249   | 3     1     469   |     | 8     246    7      | 2469    5    249    | 3       1     469   |
:-------------------+---------------------+-------------------:     :---------------------+---------------------+---------------------:
| 1235  12345  124  | 23479   1249  8     | 479   6     34579 |     | 1235  12345  1249   | 234579  129  8      | 479     6     4579  |
| 1356  7      1468 | 349     149   1349  | 2     3458  3459  |     | 1356  7      14689  | 345689  189  13     | 2       38    4569  |
| 9     2348   248  | 2347    6     5     | 478   3478  1     |     | 9     234    248    | 23478   6    5      | 478     2378  1     |
'-------------------'---------------------'-------------------'     '---------------------'---------------------'---------------------'

different to regular GSP, the resulting puzzle does not follow the rules of Sudoku in the same way, the boxes are meaningless for instance
this was why i was unsure about it. but the given digits map to themselves in the same fashion as GSP, in cycles of 1 or 2 (i forgot to mention cycles greater than 2 do not exist)

after some time milling it over, i think i've convinced myself that this is not a logical deduction, at least not in the way i'm using it. theoretically, what is to stop a cycle of 3 or 4 just by looking at the givens?
ryokousha provided a quick counter-example on discord that has given digits mapping in the same way as my puzzle, but the solution does not follow through

Code: Select all

counter-example
.....9..6.462.3...7.....1...2.8...4.....6587.867..231..............5..983.1......

as an aside, because i found it interesting, if you transform these puzzles as rZcY=X you get a puzzle with diagonal symmetry on every digit

i'm still a little hopeful someone might find something interesting with this idea or this specific puzzle, so i wont delete the post even though it now seems fatuous

[StrmCkr]

im unsure how you got that grid, for me it looks like so:

i only swapped the
RxC(y) = Z became RxC[z]=y

where z was a digit in the cell and Y was the Col sector selected.

and translated the grid using the same system for all cells.

..5.3....
4..1.....
.....6.89
....5.7..
.23.....9
1.7.5.38.
.....6.8.
.7....2..
9...65..1

edit: apparently i was manually only applying it to the first cell and not both.

[shye]

unless im missing sonething this looks like theres an error, since r6c7=3 -> r6c3=7 but you have r6c7=3 -> r6c3=3

[StrmCkr]

sym check ran to completion with none applicable

[ghfick]

The two views of the grid are the RC View and the RN View as presented by Denis Berthier in his book 'The Hidden Logic Of Sudoku'.
What is very interesting about this puzzle is that the RC view of the givens is the same as the RN View of the givens.
Maybe quite rare?

[shye]

What is very interesting about this puzzle is that the RC view of the givens is the same as the RN View of the givens.
Maybe quite rare?

60074 solutions have the property (thank you mith for running a search )

[StrmCkr]

That is fascinating didn't even think to check that as my code does it for extended grid display.



There is 4 views of a gird
RC
RN
CN
BN

EXTENDED VIEWS INCLUDE:
MINI ROW
MINI COL
BOX MINI ROW
BOX MINI COL
ERI

so the question you where originally asking to begin with is can you directly swap the pms of the RN view to the RC view since the sets are the same..


{my codes zero based and is the same as yzf's} Rn view display with marks enabled.

[eleven]

... someone might find something interesting with this idea ...

At least you can make a sudoku variant, with the additional constraint, that the solution must have this kind of symmetry

[blue]

What is very interesting about this puzzle is that the RC view of the givens is the same as the RN View of the givens.
Maybe quite rare?

60074 solutions have the property (thank you mith for running a search )

60074 is the number of ways to complete this grid:

Code: Select all

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

The total number of grids is larger by a factor of 6*6*6*6*6*6 ... 2,802,812,544.

There are 1912 "ED" (essentially distinct) classic sudoku grids, that have a transformation to a form showing the symmetry.
On the other hand, there are 6^8 transformations that preserve the symmetry, and using those transformations only, the 2,802,812,544 grids can be reduced to 1954 distinct cases.

The symmetry preserving transformations allow permuting bands and stacks and permuting rows within bands and columns within stacks, and mapping cell values in the same way as the column numbers. For example: swap columns 4 & 5, and map 4's to 5's and vica-versa.

This grid:

Code: Select all

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

... is "(row) minlex" with respect to the symmetry preserving transformations.
It's a morph of the MC ("most canonical") grid.
It has 36 automorphisms in the group of symmetry preserving transformations.
36 is the largest automorphism count, and this is the only (row minlex) grid with that count.

This grid:

Code: Select all

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

... is the "column minlex" version of the grid above -- "minlex" when the grid is read out by going down the columns, and working from left to right.
As a classic sudoku grid, it looks a little more symmetric than the "row minlex" version -- repeating minirows.

Like this one, each "column minlex" grid is an extension of the (partial) grid at the top of the post.

Cheers.

[StrmCkr]

hey
Blue
sorry for hijacking I'll delete after

deleted to remove irrelevant content

[blue]

(deleted)

[shye]

... someone might find something interesting with this idea ...

At least you can make a sudoku variant, with the additional constraint, that the solution must have this kind of symmetry

hopefully this puzzle works like i think it does! might be tricky
additional rule: digits in each cell give the position of N in the row, where N is the column number

Code: Select all

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

[eleven]

This one is really hard for me. Apart from very simple symmetry operations, which did not really make useful progress, i could not find a helpful technique. So i needed a complex net to solve it.

Have you found any effective technique ?