Fully symmetrical puzzles

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Re: Fully symmetrical puzzles

Postby champagne » Sun Apr 28, 2013 5:48 am

I made a far (depth 6) vicinity search on the first puzzles and got the following, a poor result
As Mike, I now start the search on the pattern game pattern.



....1.......2.3.....1.4.2...1.5.6.3.3.5...7.8.8.3.2.9...4.6.9.....1.8.......2....
....1.......2.3.....1.4.2...1.5.6.3.3.6...7.8.8.3.2.9...4.5.9.....1.8.......2....
....1.......2.3.....1.4.5...1.3.5.2.6.5...7.8.4.8.9.3...2.9.6.....7.4.......3....
....1.......2.3.....1.4.5...2.6.7.3.3.8...6.9.9.3.2.4...4.7.8.....5.9.......2....
....1.......2.3.....2.4.5...3.6.7.8.8.5...6.3.4.8.2.7...9.7.4.....1.8.......2....
....1.......2.3.....4.5.6...1.3.6.2.2.7...8.9.5.9.2.1...8.3.5.....1.9.......6....
....1.......2.3.....2.4.5...1.5.6.3.3.7...8.9.4.3.7.1...9.6.2.....9.1.......5....
....1.......2.3.....2.4.5...3.1.6.7.7.8...9.3.4.3.7.1...5.6.4.....8.1.......2....
....1.......2.3.....2.4.5...3.6.4.1.7.5...6.3.6.1.7.2...8.2.7.....7.1.......9....
....1.......2.3.....2.4.5...3.6.4.1.7.8...6.3.6.1.7.2...4.2.7.....7.1.......9....
....1.......2.3.....2.4.5...3.6.7.8.8.9...7.3.7.3.2.6...5.2.4.....9.8.......6....
....1.......2.3.....2.4.5...4.6.2.1.6.7...8.2.9.3.5.6...8.5.4.....1.6.......3....
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Postby Afmob » Sun Apr 28, 2013 6:53 am

Hi claudiarabia,

feel free to use them.
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Re: Fully symmetrical puzzles

Postby m_b_metcalf » Sun Apr 28, 2013 7:07 am

claudiarabia wrote:May I please use these puzzles in my Internet-Blog of course with reference to your names?

Gladly.

Mike
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Re: Fully symmetrical puzzles

Postby claudiarabia » Mon Apr 29, 2013 12:22 pm

@Afmob @M-B-Metcalf: Thank you:)
And thank you, champagne too. For now I have also a sudoku with the rating 6.6 of the SE

regards
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Re:

Postby Serg » Wed Aug 14, 2013 12:40 pm

Hi, JPF!
JPF wrote:Now, we have the number N'(n) of S-different patterns with n clues :
Code: Select all

Number            Number             Number
of clues        of patterns        of S-different
(n)               N(n)              patterns N'(n)

 0                  1                  1
 1                  1                  1
 4                  8                  4
 5                  8                  4
 8                 34                 10
 9                 34                 10
12                104                 24
13                104                 24
16                253                 52
17                253                 52
20                512                 98
21                512                 98
24                888                165
25                888                165
28               1344                246
29               1344                246
32               1794                323
33               1794                323
36               2128                380
37               2128                380
40               2252                402
41               2252                402



N'(81-k) = N'(k) ; k=0,...,81

which gives a total of 6016 S-different patterns (32768 before) ...

Could you describe your method of calculation number of patterns for n-clue essentially different fully symmetrical patterns? I see no way of using standard combinatorial enumeration techniqes, such as Burnside Lemma and Polya Enumeration Theorem, because of VPT acting on "fully symmetrical patterns" set don't form a group. I think developing enumeration method for sets having symmetry constraints (like fully symmetrical patterns set) is nice mathematical challenge.

Serg
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Re: Fully symmetrical puzzles

Postby JPF » Thu Aug 15, 2013 5:39 am

Hi Serg,
As you probably saw in my year-2006 post, the list of all the non-equivalent patterns was given by Gordon Royle.
Here attached are the files I got from him.

JPF
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Re: Fully symmetrical puzzles

Postby Serg » Thu Aug 15, 2013 12:50 pm

Hi, JPF!
Thank you for clarification. I've sent PM to Gordon about methods of e-d fully symmetrical patterns enumeration.

Serg
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Re: Fully symmetrical puzzles

Postby JPF » Sat Aug 17, 2013 5:39 am

Hi Serg!

I checked Gordon's numbers.
Actually those numbers are easy to calculate by canonalization of the potential Fully Symmetrical patterns.

So there are two steps.
Let's assume we are considering a pattern with n clues.

1. Potential fully symmetrical patterns N(n):

As explained at the beginning of the thread, the pattern is defined by the 15 cells of the upper left triangle of the grid.
We write zi = 1 if the ith cell has a clue, zi = 0 if not.
Each cell generates, by the required symmetries, t cells.
t is equal to: 8 or 4 for the cells in the main diagonal and the vertical axis, 1 for the cell in the middle of the grid.

Then, a upper left triangle must be such that: t1z1 + t2z2 +...+ t15z15 = n ; (zi = 0 or 1)
The number of solutions of this equation, easy to calculate, is N(n) in my chart.
At the same time we get the corresponding patterns.


2. Non equivalent patterns N'(n).

I used dobridchev's gridchecker -very very fast!- to eliminate the equivalent patterns.
I got the same numbers as those given by Gordon.

JPF
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Re: Fully symmetrical puzzles

Postby Serg » Sun Aug 25, 2013 3:22 pm

Hi, JPF!
Your method is clear and straightforward, but I'd like to have mathematical calculated numbers for following reasons:
1. Neither program (tool) is guaranteed not to have bugs, so it is useful to have mathematically calculated results for proving program results.
2. It would be nice to have a (mathematical) method of enumeration e-d patterns belonging to arbitrary class of patterns. It would simplify the enumeration task and could get more reliable results. For example, how many are there 18-clue vertically symmetric patterns? Counting such patterns is rather complicated task, and cannot be solved so easy as counting fully symmetric patterns. One has to write special program (I've just wrote it) to enumerate such vertically symmetric patterns. To make this program fast I have to complicate logic of this program, so I am not quite sure the program's results are correct.

I pondered possible mathematical foundation of such symmetric patterns enumeration (vertically symmetric, diagonally symmetric, etc.). I think it is necessary to generalize combinatorial enumerarion methods - instead of group of transformation acting on G-set we should consider set of subgroups of VPT group, each acting on its own set of patterns (sets can intersect). But this problem is too complicated, at least I don't know fruitful approach to this problem yet.

Serg

P.S. Gordon Royle didn't respond to my PM.
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Postby Pat » Mon Aug 26, 2013 1:23 pm

Serg wrote:P.S. Gordon Royle didn't respond to my PM.

gfroyle has not been active on the forum for several years
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