Perfectly symmetric patterns

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Re: Perfectly symmetric patterns

Postby JPF » Mon Nov 30, 2020 8:30 am

I would add these two important old threads dealing with patterns and/or geometric symmetries.

1. Basic and rigourous definitions of symmetry by Gordon Royle:
Sudoku Symmetry - Formalized
The link to his paper doesn't work, but I attached it below.

sudoku-symmetry Gordon Royle.pdf
(121.07 KiB) Downloaded 20 times

2.Sudokus with an original rare shape

Last point, if one of your goals is to find the highest rating for a given pattern, techniques developped in the Patterns Game are definitively usefull.
For instance:

Code: Select all
 . . 1 | 2 . 3 | 4 . .
 . 5 . | . . . | . 2 .
 4 . . | . . . | . . 6
-------+-------+-------
 1 . . | . 4 . | . . 7
 . . . | 6 . 1 | . . .
 8 . . | . 9 . | . . 5
-------+-------+-------
 9 . . | . . . | . . 4
 . 3 . | . . . | . 6 .
 . . 4 | 7 . 9 | 8 . .    SER=9.9

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Re: Perfectly symmetric patterns

Postby m_b_metcalf » Mon Nov 30, 2020 8:32 am

denis_berthier wrote:
m_b_metcalf wrote:You could check on all the played patterns (obtainable on demand with a
Code: Select all
::: patterns :::
post).

I tried this, but it doesn't seem to work (or maybe it takes time for getting an answer?)

It seems difficult to use the Patterns Game thread. It has 46,000 posts and probably several thousand patterns. Is there any way to search it for some specific pattern?

Yes, sometimes it takes a while to respond. You're looking at the wrong thread - you can very easily inspect the Results thread where you see the patterns and the results in a compact form. To search for a particular pattern you need the patterns file. I append a version I have. Note: the patterns are in the right-hand half of the list.

HTH

Mike
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patterns.txt
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Re: Perfectly symmetric patterns

Postby Serg » Mon Nov 30, 2020 5:50 pm

Hi, Denis!
denis_berthier wrote:
Serg wrote:Such patterns were discussed in the thread Fully symmetrical invalid patterns in 2017. Two lists of "basic" patterns were published there - 25 invalid patterns hot having valid puzzles and 49 valid patterns having valid puzzles.

This will be of great help for the present thread and I'll use this before publishing more about the other Families.
To be more precise, skipping the beginning of the thread, the main results are here: http://forum.enjoysudoku.com/fully-symmetrical-invalid-patterns-t33569-53.html and in the next 4 or 5 posts. Can you confirm that those are the final lists of valid and invalid fully-symmetrical patterns?

My post dated by "Sun Mar 12, 2017 8:32 pm" (the topmost post at the page http://forum.enjoysudoku.com/fully-symmetrical-invalid-patterns-t33569-53.html) contains attached files "fullsym_patterns_invalid.zip" (full list of 871 essentially different invalid fully symmetrical patterns) and "fullsym_patterns_valid.zip (full list of 5145 essentially different valid fully symmetrical patterns, in total - 6016 ed fully symmetrical patterns). Those lists are final.
denis_berthier wrote:
Serg wrote:If any fully symmetrical pattern is subset of some "basic" invalid pattern, it has no valid puzzles. If any fully symmetrical pattern is superset of "basic" valid patterns, it has at least one valid puzzle.

When you say "it has valid puzzles", do you mean "not necessarily minimal ones", "not necessarily symmetrically-minimal ones" or merely (as I understand it) valid ones (i.e.single-solution)?
Can you confirm that the above referenced patterns are resp. maximal/minimal-symmetric?

When I am saying "(some) pattern has valid puzzles" I assume "not necessarily minimal valid puzzles". Valid puzzles, as usual, must have unique solution. Published lists of 25 symmetrically maximal fully symmetrical invalid patterns are symmetrically maximal in the sense that addition of any clues preserving full symmetry makes those patterns valid. Any fully symmetrical invalid pattern must be subset of some pattern(s) from this list. But all 25 patterns are independent, i.e. the aren't subsets of each other. Similarly the list of 49 symmetrically minimal fully symmetrical valid patterns becomes invalid by removal of any clues preserving their full symmetry.

Some of symmetrically maximal invalid patterns are maximal in general sense, i.e. they becomes valid by adding any one clue (not necessarily preserving full symmetry). For example, well-known
Code: Select all
  Magic Pattern

x x x x x x x x x
x x x . . . x x x
x x x . . . x x x
x . . . . . . . x
x . . . x . . . x
x . . . . . . . x
x x x . . . x x x
x x x . . . x x x
x x x x x x x x x

is maximal in general sense.

Saying about minimality of valid puzzles produced by symmetrically minimal valid patterns. When those patterns are minimal in general sense, i.e. they become invalid by removing any one clue, they have minimal valid puzzles only. More detailed it's discussed in the thread Minimal patterns.

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Re: Perfectly symmetric patterns

Postby denis_berthier » Tue Dec 01, 2020 9:32 am

Serg, thanks for these explanations.
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Re: Perfectly symmetric patterns

Postby denis_berthier » Wed Dec 02, 2020 6:28 am

m_b_metcalf wrote:
denis_berthier wrote:
m_b_metcalf wrote:You could check on all the played patterns (obtainable on demand with a
Code: Select all
::: patterns :::
post).

I tried this, but it doesn't seem to work (or maybe it takes time for getting an answer?)

Hi Mike,
It seems difficult to use the Patterns Game thread. It has 46,000 posts and probably several thousand patterns. Is there any way to search it for some specific pattern?

Yes, sometimes it takes a while to respond.


Hi Mike
I finally got the response. I'm surprised to see that:
- there are only 435 patterns; can the same pattern be re-used for several games?
- the highest SER is 9.2; isn't the game about finding high SER puzzles?
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Re: Perfectly symmetric patterns

Postby JPF » Wed Dec 02, 2020 8:24 am

As defined, the "slashes root pattern" and the "wawes root pattern" are equivalent.

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Re: Perfectly symmetric patterns

Postby denis_berthier » Wed Dec 02, 2020 8:28 am

JPF wrote:As defined, the "slashes root pattern" and the "wawes root pattern" are equivalent.

You're right, but not all equivalent patterns make visually interesting patterns.

Is this an explanation for the small number of patterns in the Patterns Game: equivalent patterns count for one?
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Re: Perfectly symmetric patterns

Postby m_b_metcalf » Wed Dec 02, 2020 8:46 am

denis_berthier wrote:Hi Mike
I finally got the response. I'm surprised to see that:
- there are only 435 patterns; can the same pattern be re-used for several games?
- the highest SER is 9.2; isn't the game about finding high SER puzzles?

At the very beginning it was possible to reuse a pattern but that is no longer possible (nor its equivalent).
The ratings in the patterns list are those of the dealer which, apart from those at the very beginning, have to be solvable with singles only. For actual played ratings you need to read the Results thread. The highest ratings per clue count are given below. I recommend also that you read through this.

HTH

Mike

P.S. the first 30 or so games are unnumbered.

Code: Select all
  #   ER   EP   ED    Game         Dealer       Player
clues
 19  10.4 10.4  3.4    381         coloin  m_b_metcalf
 20  11.8 11.8  3.4    169    m_b_metcalf          gsf
 21  11.8 11.8 10.7    349    m_b_metcalf     1to9only  (overalll highest)
 22  11.7  1.2  1.2    193         coloin    champagne
 23  11.4 11.4 10.8    148       Mauricio          gsf
 24  11.3 11.3 10.6    209      champagne  m_b_metcalf  (also in Game 212)
 25  11.3 11.3 11.1    309        Patrice      papy999
 26  11.1 11.1 10.4    302        Patrice      papy999
 27  10.7 10.7  8.9    150    m_b_metcalf       eleven
 28  10.5 10.5  9.2    155       Mauricio    champagne
 29  10.4 10.4  9.9    375    m_b_metcalf          jpf
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Re: Perfectly symmetric patterns

Postby denis_berthier » Wed Dec 02, 2020 9:28 am

Mike, thanks
It's all clear now.
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