Symmetric 18s

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

Re:

Postby Serg » Thu Dec 12, 2013 3:06 pm

Hi, Afmob!
Afmob wrote:Your puzzle #120A is not valid.

You are right. U4 Ua set cannot occupy 4 boxes. Therefore puzzle #120A was produced incorrectly.

But sudoku superpuzzle remains true (if pattern #120 has really 1 valid puzzle only).

Serg
Serg
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Re: Symmetric 18s

Postby eleven » Thu Dec 12, 2013 4:35 pm

Since there are so many equivalence transformations of a pattern, to each many nice presentations can be found.
E.g. this is a small collection for pattern #120:
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Code: Select all
+-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+
| . . . . x . . . . |  | . . . . x . . . . |  | . . . . . . . . . |  | . . . x . x . . . |  | . . . . x . . . . |  | . . . . . . . . . |  | . x . . . . . x . |    | . . . . . . . . . |
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+-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+    +-------------------+
+-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+
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+-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+    +-------------------+
+-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+
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| x . . . . . . . x |  | . . . x x x . . . |  | . . . x . x . . . |  | . x . . x . . x . |  | . . . x x x . . . |  | . x . . x . . x . |  | . x . . x . . x . |    | x . . . x . . . x |
+-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+  +-------------------+    +-------------------+
eleven
 
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Re:

Postby Serg » Thu Dec 12, 2013 9:29 pm

Hi, Afmob!
Afmob wrote:Edit: Here is another new pattern:
Code: Select all
...........1...2...3..4..5.....3......21.67..8.......1...2.3......7.1....5.....8.

Congratulations on finding pattern #121! I confirm it is new. Well done!
Please, post new puzzles (patterns) in separate posts.

Serg
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Re: Symmetric 18s

Postby Serg » Fri Dec 13, 2013 5:03 am

Hi, eleven!
eleven wrote:Since there are so many equivalence transformations of a pattern, to each many nice presentations can be found.
E.g. this is a small collection for pattern #120:

Nice patterns! It would be interesting to try searching for more symmetric configuration, more compact or less compact configurations, etc. For example, one can establish "compaction measure" - the sum of square distances between clue cells and centre of grid. Clue r3c2 would have "compaction measure" = 13 (3^2+2^2). So, we could search for patterns having minimal "compaction measure" (sum done by all clues). Or another idea - calculate "symmetry measure" by finding all 4 possible reflections (around 4 symmetry axis) for each clue, and counting those reflections which coincide with real clues. Then such sum should be divided by 4 x 18 (each clue can be counted at most 4 times), and we'll get positive value being not greater than "1". Fully symmetric (dihedral symmetric) patterns would have exact "1" for this measure, asymmetric patterns would have measure values close to zero.

Serg
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Re: Symmetric 18s

Postby eleven » Fri Dec 13, 2013 12:46 pm

I tried it to get a maximum of symmetric and connected pattern cells (where for the same value i took the one with minimum distances to the center). However this can't give my favorite shapes of course.
See https://sites.google.com/site/sudoeleven/v18maxSym.jpg and https://sites.google.com/site/sudoeleven/v18maxConnect.jpg. Patterns are in the order they were posted.
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Re: Symmetric 18s

Postby Serg » Mon Dec 16, 2013 10:58 pm

Hi, eleven!
eleven wrote:I tried it to get a maximum of symmetric and connected pattern cells (where for the same value i took the one with minimum distances to the center). However this can't give my favorite shapes of course.
See https://sites.google.com/site/sudoeleven/v18maxSym.jpg and https://sites.google.com/site/sudoeleven/v18maxConnect.jpg. Patterns are in the order they were posted.

I am little confused by your post.
Let's consider patterns #120 in both your collections. Pattern #120 in the file "v18maxSym.jpg" has not maximal possible (for this pattern) symmetry metric. Pattern #120 in the file "v18maxConnect.jpg" has nothing common with possible configurations produced by "minimal sum of square distances to the centre of grid". But instead it introduced new interesting pattern selection criterium - "select connected isomorphs, i.e. patterns, having such clue configuration, that Chess King can rich every clue moving through clues only."

Please, describe your selection criteria for both files more detailed.

Serg
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Re: Symmetric 18s

Postby eleven » Tue Dec 17, 2013 10:00 pm

Serg,
please note, that these two collections only were quick tries.
For the first i gave 2 points for a pattern cell in the upper half, which had a mirrored cell in the lower half, and one for pattern cells in row 5.
For the second i gave one point for each neighboured cell diagonally or vertically downward.
Out of the equivalents with most potints i took the first one with minimum sum of distances to the center.

Though the results were not bad, i am not satisfied with them.

What i would like to find are more significant representatives for each pattern, which e.g. could be be seen as flowers, smileys, mushrooms, huts or whatever (some of them in groups, some singular).
The best method i found for that so far, was a little pogram,that shows 10x10 shapes of a pattern, where all 100 ms from top left to bottom right one was changed randomly. Then i could click on each pattern i liked, to save it. However after doing that for 3 patterns my eyes hurt :)

I am short of time until the christmas days, maybe later, when afmob hopefully will have finished his work, i will get back to this.
eleven
 
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Re: Symmetric 18s

Postby Serg » Wed Dec 18, 2013 8:03 pm

Hi, eleven!
My previous post was too brief. So, I decided to discuss "symmetry metric" theme more detailed.

Let's consider "symmetry metric", which was proposed by me to evaluate sudoku puzzles "aesthetic pleasure" (i.e. to introduce some mathematical criterium - how to select the most proper isomorphic view of new found pattern?).

I proposed calculate symmetry metric by finding all 4 possible reflections (around 4 symmetry axis) for each clue, and counting those reflections which coincide with real clues. Then such sum should be divided by 4 x 18 (each clue can be counted at most 4 times), and we'll get positive value being not greater than "1". It is evident that this metric preserved, at least, when every pattern transformed in such ways:

1. Pattern is reflected around vertical symmetry axe.
2. Pattern is reflected around horisontal symmetry axe.
3. Pattern is rotated by 90/180/270 degrees.

Let's consider symmetric 18-clue pattern posted by Afmob (I replaced puzzle's numbers by "1" to distinguish pattern):
Code: Select all
   Pattern #120

. . . . . . . . .
. . 1 . . . 1 . .
. 1 . . 1 . . 1 .
. . . . 1 . . . .
. . . 1 1 1 . . .
1 . . . . . . . 1
. . . 1 . 1 . . .
1 . . . . . . . 1
1 . . . 1 . . . 1

Let's calculate symmetry metric for this pattern. Cell r2c3 has 2 symmetric reflections - cells r2c7 and r3c2, so it must be counted with "2" weight, central cell r5c5 has 4 "autoreflections" (it coincides with its own reflections), so it must be counted with "4" weight, etc. Doing loop over all clue cells of this pattern and summing results, we'll get symmetric metric 33 (or 33/72 = 46 % approx.).

Let's consider isomorph of pattern #120 posted by you (file "v18maxSym.jpg"):
Code: Select all
   Pattern #120a

. . . . 1 . . . .
. . 1 . . . 1 . .
. . . 1 1 1 . . .
. 1 . . . . . 1 .
1 . . . 1 . . . 1
. . . . . . . . .
. . . 1 . 1 . . .
. . 1 . . . 1 . .
. . 1 . 1 . 1 . .

If my manual calculations were not wrong, symmetry metric for pattern #120a is 41 (57 %), somewhat higher than symmetry metric for pattern #120. But there are isomorphs of #120 having higher symmetry metric. I consider all possible isomorhs of the pattern #120 and found 12 patterns (unique up to reflections around horisontal symmetry axe, main diagonal, antidiagonal and to rotations by 90, 180 and 270 degrees), having maximal possible symmetry metric - 57 (79 %). Here they are:
Code: Select all
      #120b                 #120c                 #120d                 #120e

. . . . 1 . . . .     . . . . 1 . . . .     . . . . 1 . . . .     . . . . 1 . . . .
. . . 1 1 1 . . .     . . . 1 1 1 . . .     . 1 . . . . . 1 .     . 1 . . . . . 1 .
. . 1 . . . 1 . .     . . 1 . . . 1 . .     . . . 1 1 1 . . .     . . . 1 1 1 . . .
. . . . . . . . .     . 1 . . . . . 1 .     . . . . . . . . .     . . 1 . . . 1 . .
1 . . . 1 . . . 1     1 . . . 1 . . . 1     1 . . . 1 . . . 1     1 . . . 1 . . . 1
. 1 . . . . . 1 .     . . . . . . . . .     . . 1 . . . 1 . .     . . . . . . . . .
. . 1 . . . 1 . .     . . 1 . . . 1 . .     . . . 1 . 1 . . .     . . . 1 . 1 . . .
. . . 1 . 1 . . .     . . . 1 . 1 . . .     . 1 . . . . . 1 .     . 1 . . . . . 1 .
. . 1 . 1 . 1 . .     . . 1 . 1 . 1 . .     . 1 . . 1 . . 1 .     . 1 . . 1 . . 1 .

      #120f                 #120g                 #120h                 #120i

. . . 1 1 1 . . .     . . . 1 1 1 . . .     1 . . . . . . . 1     1 . . . . . . . 1
. . . . 1 . . . .     . . . . 1 . . . .     . . . 1 1 1 . . .     . . . 1 1 1 . . .
. . 1 . . . 1 . .     . . 1 . . . 1 . .     . . . . 1 . . . .     . . . . 1 . . . .
. . . . . . . . .     1 . . . . . . . 1     . . . . . . . . .     . 1 . . . . . 1 .
. 1 . . 1 . . 1 .     . 1 . . 1 . . 1 .     . . 1 . 1 . 1 . .     . . 1 . 1 . 1 . .
1 . . . . . . . 1     . . . . . . . . .     . 1 . . . . . 1 .     . . . . . . . . .
. . 1 . . . 1 . .     . . 1 . . . 1 . .     1 . . . 1 . . . 1     1 . . . 1 . . . 1
. . 1 . 1 . 1 . .     . . 1 . 1 . 1 . .     . . . 1 . 1 . . .     . . . 1 . 1 . . .
. . . 1 . 1 . . .     . . . 1 . 1 . . .     1 . . . . . . . 1     1 . . . . . . . 1

      #120j                 #120k                 #120m                 #120n

. . . 1 1 1 . . .     . . . 1 1 1 . . .     1 . . . . . . . 1     1 . . . . . . . 1
. 1 . . . . . 1 .     . 1 . . . . . 1 .     . . . . 1 . . . .     . . . . 1 . . . .
. . . . 1 . . . .     . . . . 1 . . . .     . . . 1 1 1 . . .     . . . 1 1 1 . . .
. . . . . . . . .     1 . . . . . . . 1     . . . . . . . . .     . . 1 . . . 1 . .
. . 1 . 1 . 1 . .     . . 1 . 1 . 1 . .     . 1 . . 1 . . 1 .     . 1 . . 1 . . 1 .
1 . . . . . . . 1     . . . . . . . . .     . . 1 . . . 1 . .     . . . . . . . . .
. 1 . . 1 . . 1 .     . 1 . . 1 . . 1 .     . . . 1 . 1 . . .     . . . 1 . 1 . . .
. 1 . . . . . 1 .     . 1 . . . . . 1 .     1 . . . 1 . . . 1     1 . . . 1 . . . 1
. . . 1 . 1 . . .     . . . 1 . 1 . . .     1 . . . . . . . 1     1 . . . . . . . 1

As you can see all these patterns can be grouped by pairs (#120b and #120c, #120d and #120e, etc.), where the second pattern of the pair can be produced from the first by swapping rows r4/r6. The second curious observation is that pattern #120m contains a little copy of pattern #120b as subset, the same can be stated about patterns #120n and 120c, #120h and #120f, #120i and #120g. To my mind the finest are #120c ("Little house"), #120f (but rotated by 180 degrees - "Flower"), #120k ("Jellyfish").

Another trial to get "mechanically" nice patterns was using "compaction" criterium - "minimal sum of distance^2 to grid centre for all clue cells".

I found 8 patterns (unique up to reflections around horisontal symmetry axe, main diagonal, antidiagonal and to rotations by 90, 180 and 270 degrees), having minimal possible compaction metric. But some of them have very similar "brother" patterns not having vertical symmetry. For example:
Code: Select all
      #120p                 #120q
. . . . . . . . .     . . . . . . . . .
1 . . . . . . . 1     1 . . . . . . 1 .
. 1 . . 1 . . 1 .     . 1 . . 1 . . . 1
. . . 1 . 1 . . .     . . . 1 . 1 . . .
. . 1 . 1 . 1 . .     . . 1 . 1 . 1 . .
. . 1 . . . 1 . .     . . 1 . . . 1 . .
. . . 1 1 1 . . .     . . . 1 1 1 . . .
. . 1 . . . 1 . .     . . 1 . . . 1 . .
. . . . 1 . . . .     . . . . 1 . . . .

Sums of the squre distances for cells r2c9, r3c8 (#120p) and cells r2c8, r3c9 (#120q) are equal to each other and this is problem for compaction criterium. To avoid this problem I used distance^4 instead of distance^2. This helps to filter out extra similar patterns.

Finally (using distance^4) I got 2 patterns, having minimal possible compaction metric. Here they are:
Code: Select all
      #120r                 #120s
. . . . . . . . .     . . . . . . . . .
. 1 . . . . . 1 .     . 1 . . . . . 1 .
1 . . . 1 . . . 1     1 . . . 1 . . . 1
. . . 1 . 1 . . .     . . 1 . . . 1 . .
. . 1 . 1 . 1 . .     . . 1 . 1 . 1 . .
. . 1 . . . 1 . .     . . . 1 . 1 . . .
. . . 1 1 1 . . .     . . . 1 1 1 . . .
. . 1 . . . 1 . .     . . 1 . . . 1 . .
. . . . 1 . . . .     . . . . 1 . . . .

Rather nice patterns. Again the second can be produced from the first by swapping rows r4/r6.

Serg

[Edited. I realized the reason why swapping rows r4/r6 didn't change "symmetry metric" and "compaction metric".
Symmetry metric will not be changed if we'll swap rows r4/r6 and columns c4/c6 (it is rather evident). But swapping columns c4/c6 doesn't change vertically symmetric patterns. So, for vertically symmetric patterns swapping rows r4/r6 alone doesn't change pattern too.
One can easily notice that distance of any cell to central cell doesn't get changed after swapping rows r4/r6 or after swapping columns c4/c6.
So, it is not curious that all symmetric patterns published in my post didn't changed after swapping rows r4/r6.]

[Edited2. I corrected errors in patterns #120r and #120s (manual errors during copying patterns).]
Last edited by Serg on Fri Dec 20, 2013 8:32 am, edited 2 times in total.
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Re: Symmetric 18s

Postby dobrichev » Thu Dec 19, 2013 7:00 am

The first things that come in mind for investigating further formal symmetry of the morphs are
- Center of mass = sum(row(i)). On x axis It lies by definition on c5, but on y axis it could be moved farther/closer to r5.
- Second moment of area = sum(dist^2). It could be examined separately over x, y, and say diagonal axis.
- Moment of inertia over the center of mass or over r5c5 or over r9c5 = sum(dist^2). Minimal and maximal values could be examined, i.e. moving the givens as close/far as possible to/from the chosen "center".

Another approach is to examine the topology, possibly determining equivalence classes. OCR algorithms could help there.
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Re: Symmetric 18s

Postby Serg » Thu Dec 19, 2013 7:35 am

Hi, Mladen!
dobrichev wrote:The first things that come in mind for investigating further formal symmetry of the morphs are
- Center of mass = sum(row(i)). On x axis It lies by definition on c5, but on y axis it could be moved farther/closer to r5.
- Second moment of area = sum(dist^2). It could be examined separately over x, y, and say diagonal axis.
- Moment of inertia over the center of mass or over r5c5 or over r9c5 = sum(dist^2). Minimal and maximal values could be examined, i.e. moving the givens as close/far as possible to/from the chosen "center".

Another approach is to examine the topology, possibly determining equivalence classes. OCR algorithms could help there.

Thank you for ideas for further investigations. I feel this area can absorb my spare time for years.
I'll check moments calculation coming days.

Serg
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Re: Symmetric 18s

Postby Serg » Thu Dec 19, 2013 8:03 am

Hi, people!
I realized the reason why swapping rows r4/r6 didn't change "symmetry metric" and "compaction metric".
Symmetry metric will not be changed if we'll swap rows r4/r6 and columns c4/c6 (it is rather evident). But swapping columns c4/c6 doesn't change vertically symmetric patterns. So, for vertically symmetric patterns swapping rows r4/r6 alone doesn't change pattern too.

One can easily notice that distance of any cell to central cell doesn't get changed after swapping rows r4/r6 or after swapping columns c4/c6.

So, it is not curious that all symmetric patterns published in my post didn't changed after swapping rows r4/r6.

Serg
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Re: Symmetric 18s

Postby ronk » Thu Dec 19, 2013 4:09 pm

For the topic of vertically symmetric 18s, I vote for continued use of row minlex patterns.

It's an unbiased method that is simple to use, even manually.
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Re: Symmetric 18s

Postby Serg » Fri Dec 20, 2013 8:41 am

Hi, ronk!
ronk wrote:For the topic of vertically symmetric 18s, I vote for continued use of row minlex patterns.

It's an unbiased method that is simple to use, even manually.

Please, point me at least one post in this thread, where published puzzle was represented in minlex form. Instead, all puzzles were published in symmetric form to demonstrate their symmetry and aesthetic properties. I am only trying to find unique symmetric form which would preserve additional symmetries (if any) and produce pleasant for eye view.

Serg
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Re: Symmetric 18s

Postby eleven » Fri Dec 20, 2013 11:15 am

In the process of finding new patterns the minlex form, suggested by Ron 3 years ago here was the most practicable for me, similar to the minlex normalization (canonicalization) of puzzles. It was easy to determine, if a pattern is new, and also to look through the patterns for special properties.
However this process (hopefully) is coming to an end now, and we will know all possible patterns.
So it is another thing to think about other presentations, which might look just prettier.
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Re: Symmetric 18s

Postby ronk » Sat Dec 21, 2013 12:36 am

Serg wrote:
ronk wrote:For the topic of vertically symmetric 18s, I vote for continued use of row minlex patterns.

It's an unbiased method that is simple to use, even manually.

Please, point me at least one post in this thread, where published puzzle was represented in minlex form. Instead, all puzzles were published in symmetric form to demonstrate their symmetry and aesthetic properties.

Yes, sorry, my statement makes a sloppily written definition, so let me try again.

Given a puzzle with vertical symmetry, choose the vertically symmetric morph of the clue pattern with row-order minimal lexicographical order.
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