## Down Under Upside Down - a Sudoku puzzle

Post puzzles for others to solve here.
eleven wrote:
udosuk wrote:On the other hand, I have this approach (which I haven't confirmed to work yet)...

Interesting, but from my feeling i cant believe, that it works. I guess, that partial symmetry does not say anything about the rest of the grid.

After a bit of studying I have to agree with you, that it's probably not too useful. Back to the drawing board...

Meanwhile here is another isomorph, which brings the empty block to the centre, and makes the whole clue pattern a lot more "symmetrical". Also I decided to swap the digits 2 & 5 so that the digit cycles are now {987|654|321}:

Code: Select all
`+-------+-------+-------+| 9 . . | 3 . . | . . 7 || . 8 . | . 2 . | 9 . . || . . 7 | . . 1 | . 8 . |+-------+-------+-------+| 3 . . | . . . | 4 . . || . 2 . | . . . | . 6 . || . . 1 | . . . | . . 5 |+-------+-------+-------+| . 9 . | 6 . . | 3 . . || . . 8 | . 5 . | . 2 . || 7 . . | . . 4 | . . 1 |+-------+-------+-------+`
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:I guess that makes the 7th main type of automorphism. Perhaps we could call it "mini-diagonal" (although it also includes "mini-broken-diagonals" )...

Interesting. I did more studying and find out that there might be an "8th type" of automorphism. No time to make a puzzle but here is an example solution grid:

Code: Select all
`+-------+-------+-------+| 1 4 7 | 3 5 9 | 2 6 8 || 2 5 8 | 1 6 7 | 3 4 9 || 3 6 9 | 2 4 8 | 1 5 7 |+-------+-------+-------+| 8 1 6 | 4 7 2 | 5 9 3 || 9 2 4 | 5 8 3 | 6 7 1 || 7 3 5 | 6 9 1 | 4 8 2 |+-------+-------+-------+| 5 8 1 | 9 3 4 | 7 2 6 || 6 9 2 | 7 1 5 | 8 3 4 || 4 7 3 | 8 2 6 | 9 1 5 |+-------+-------+-------+`

You can see each mini-column is either {123|456|789} with the same cyclic order. So the mapping [r231564897]+[d123456789->d312645978] should restore the grid. However I don't see this automorphism to be much powerful/helpful in solving. I would call it "mini-line" symmetry (because you can have mini-row or mini-column with it).

Is that all? Or will there be a "9th type"?
udosuk

Posts: 2698
Joined: 17 July 2005

I made some search now and (re)found this thread Sudoku Symmetry - Formalized, where you also contributed
On page 3 there is a link to Red Eds symmetry group, where all possible automorphisms are listed. Yours should be class 8.

If i counted right, then there are 26 non-trivial classes, where grids are possible. Its neither easy to find our well-known symmetries nor to see, what the simplest transformations are to get those automorphisms.
Some seem to be very exotic with only a few grids, which can have them (like class 23 with 162 possible grids).
eleven

Posts: 2270
Joined: 10 February 2008

Thanks for the reminder for my bad memory.

Will study Red Ed's great web page about symmetry group. I see that many of the 275 classes are combination of 2 or more "automorphisms" we were talking about. But I find it hard to believe there are as many as 26 "simple" ones. I thought 9 or 10 was big enough.
udosuk

Posts: 2698
Joined: 17 July 2005

Note that for most classes no valid sudoku grids are possible, as you can see in column 4. It also says, how common the symmetry is. As RedEd mentioned, 180° symmetry (class 79 - 155492352 invariants) is 5 times more common than diagonal symmetry (class 37 - 30258432 invariants).

I dont have the time now to go through the 26 interesting classes. Some are obviously overlapping, since 90° symmetry (class 86) also has 180° symmetry.

From the solving POV i think, that the 2 classes we recently had here (classes 10 and 8) cant give us additional solving information like the techniques, which were used there and here for others. All you know is, that when you can place/eliminate a number you can do the same for the others of the number cycle.

It would be nice to have an overview of all (more or less useful) symmetries (with sample puzzles), but i will not be able to work on it before Christmas.
eleven

Posts: 2270
Joined: 10 February 2008

eleven wrote:I dont have the time now to go through the 26 interesting classes. Some are obviously overlapping, since 90° symmetry (class 86) also has 180° symmetry.

From the solving POV i think, that the 2 classes we recently had here (classes 10 and 8) cant give us additional solving information like the techniques, which were used there and here for others. All you know is, that when you can place/eliminate a number you can do the same for the others of the number cycle.

It would be nice to have an overview of all (more or less useful) symmetries (with sample puzzles), but i will not be able to work on it before Christmas.

From the solving POV, it seems to me that the number of classes is more or less reduced to :

central symmetry
diagonal symmetry
vertical or horizontal symmetry
.

I forget for the time being that discussion
http://forum.enjoysudoku.com/viewtopic.php?t=6459
just because I did not catch a word but may-be lack of interest because it seemed to me to be to sophisticated.

I found in recent discussions examples of diagonal and central symmetry, not of vertical or horizontal.

I introduced that symmetry analysis in my solver (sorry eleven, but I promise I will not intefere in your examples study)

If other classes gives new clues for solving actions, I did not find which ones.

For the time being, the rule is :

if you can "pair" the given digit in a "global symmetry" of givens, the paired digits can not be on the symmetry "axis" or "center".
champagne
2017 Supporter

Posts: 7049
Joined: 02 August 2007
Location: France Brittany

champagne wrote:it seems to me that the number of classes is more or less reduced to :

central symmetry
diagonal symmetry
vertical or horizontal symmetry
.

Central symmetry is not a term used by anyone else on this forum.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

ronk wrote:
champagne wrote:it seems to me that the number of classes is more or less reduced to :

central symmetry
diagonal symmetry
vertical or horizontal symmetry
.

Central symmetry is not a term used by anyone else on this forum.

you are surely right ronk, but I hope nobody has a problem with the fact that in a 180° rotation the central cell r5c5 plays a key role

Focusing on action I found the "central symmetry" more descriptive of what can be done.

Vertical symmetry -> action on column 5
Horizontal symmetry -> action on row 5
I hope I don't mix up these 2 cases toward to the standard definition

Diagonal symmetry -> action on the main (or the reverse) diagonal
Central symmetry(180° rotation) -> action on r5c5

If the cell(s) is (are) not entirely given.

champagne
champagne
2017 Supporter

Posts: 7049
Joined: 02 August 2007
Location: France Brittany

champagne wrote:Focusing on action I found the "central symmetry" more descriptive of what can be done.

I disagree ... and think rotational symmetry is much more descriptive than central symmetry. Besides, the rotational symmetry term is well established.

On the other hand, I don't share your proclivity to coin new terms.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

ronk wrote:I disagree ... and think rotational symmetry is much more descriptive than central symmetry.

If you specify 180 rotational symmetry I have no objection to your statement.

Anyway I see no reason to fight on such questions. I am more intereted in getting samples of Vertical or Horizontal symmetry.
champagne
2017 Supporter

Posts: 7049
Joined: 02 August 2007
Location: France Brittany

champagne wrote:Anyway I see no reason to fight on such questions. I am more intereted in getting samples of Vertical or Horizontal symmetry.
If you mean with vertical symmetry, that the puzzle is mirrored by column 5 (with some renumbering) to get an automorphic puzzle, this is not possible.
It would mean, that the numbers in column 5 stay unchanged, but there must be all 9 numbers (in the solution grid), so each must map to itself. But then you would have all other numbers twice in each row.
eleven

Posts: 2270
Joined: 10 February 2008

eleven wrote:
champagne wrote:Anyway I see no reason to fight on such questions. I am more intereted in getting samples of Vertical or Horizontal symmetry.
If you mean with vertical symmetry, that the puzzle is mirrored by column 5 (with some renumbering) to get an automorphic puzzle, this is not possible.
It would mean, that the numbers in column 5 stay unchanged, but there must be all 9 numbers (in the solution grid), so each must map to itself. But then you would have all other numbers twice in each row.

Quite clear. Something I should know if I had tried generation of puzzles ... or if I had spent more time in thinking.

Some code I can erase immediately. Thanks a lot.

champagne
champagne
2017 Supporter

Posts: 7049
Joined: 02 August 2007
Location: France Brittany

champagne wrote:From the solving POV, it seems to me that the number of classes is more or less reduced to :

central symmetry
diagonal symmetry
vertical or horizontal symmetry
.

There are many more (even just those helpful for solving).

The ones mentioned (so far) include:

With 1 fixed cell: 180 degree rotation, 90 degree rotation
With 9 fixed cells: diagonal reflection, "sticks"
With 0 fixed cell: whole band/stack, whole block, mini-diagonal, mini-line

http://forum.enjoysudoku.com/viewtopic.php?t=5588

Exercise 1 (ER 10.0) is a nice puzzle to try.
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:

http://forum.enjoysudoku.com/viewtopic.php?t=5588

Exercise 1 (ER 10.0) is a nice puzzle to try.

http://forum.enjoysudoku.com/viewtopic.php?t=6459

What I like in the first list (after having erased vertical and horizontal) is the very simple handling of the situation.

The most difficult is to recover the symmetry in a scrambled puzzle.

champagne
champagne
2017 Supporter

Posts: 7049
Joined: 02 August 2007
Location: France Brittany

Make sure you read this very thread carefully too, starting from p.1.

http://forum.enjoysudoku.com/viewtopic.php?t=6459

It might help a bit, but mind you, in that thread (Glyn's riddle) the situation is very different, as the puzzle on offer doesn't have symmetry in the strict sense, but we can just work out from Glyn's cryptic hints how some of the cell values in the solution grid display partial symmetry.

champagne wrote:The most difficult is to recover the symmetry in a scrambled puzzle.

On p.3, I wrote:Well, it's certainly a skill that needs a lot of practice, and an exceptional observation power.

Perhaps later I'll demonstrate a little to you in another thread.
udosuk

Posts: 2698
Joined: 17 July 2005

PreviousNext