## Symmetric 18s

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

### Re: Symmetric 18s

eleven wrote:I dont know a tool, which quickly checks, if 2 puzzles are within {-n+n} (i used gsf's for the hamming distance).

Hmm, if we could guarantee a morph for minimum Hamming distance also yielded the minimum 'n' for {-n+n}, it would be a simple revision to the counting method, which gsf would likely implement if asked.

[edit: Oops, complication #1: The quantity of clues might differ, making need for {-m+n}.]
Last edited by ronk on Fri Jun 01, 2012 1:17 pm, edited 1 time in total.
ronk
2012 Supporter

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

### Re: Symmetric 18s

Two puzzles P1 and P2 are at {-n+n} (or closer) if there exist canonical representation of (P1 {-n}) that matches the canonical representations of any of (P2 {-n}).
It is pretty fast.
dobrichev
2016 Supporter

Posts: 1621
Joined: 24 May 2010

### Re: Symmetric 18s

Nice idea, if you have a fast canonicalizer.
When i will have some time again, i want to try it using your code.
eleven

Posts: 1873
Joined: 10 February 2008

### Re: Symmetric 18s

eleven wrote:The magic number is reached, 108=2*2*3*3*3 patterns for 18=2*3*3 clue puzzles with 2 automorphisms.

Using "-f%#aC" with gsf's software results in "1" for all 108 puzzles. The result is 648 for MC ("most canonical grid"), so "%#aC" appears to be correct.

There are 64 patterns with 4 automorphs, so are you referring to the 2 "degrees of freedom" for these patterns with 4 automorphs? If so, our counts don't agree.
ronk
2012 Supporter

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

### Re: Symmetric 18s

I dont understand, what you mean. Maybe you talk of 64 patterns with exactly 2 automorphisms, but i just meant, that all these puzzles have at least 2 automorphisms, simply because they are vertically symmetric. Obviously some patterns have an additional automorphism, because you can switch rows, but i did not count them.
eleven

Posts: 1873
Joined: 10 February 2008

### Re: Symmetric 18s

Congratulations, eleven!
eleven wrote:
Code: Select all
` +-------+-------+-------+ | . . . | . 1 . | . . . | | . . 2 | . . . | 3 . . | | . . 4 | . 5 . | 6 . . | +-------+-------+-------+ | . . . | 6 . 3 | . . . | | . . 6 | . . . | 2 . . | | . 7 . | . . . | . 8 . | +-------+-------+-------+ | . . . | 4 . 2 | . . . | | . 9 . | . . . | . 1 . | | 5 . . | . . . | . . 7 | +-------+-------+-------+....1......2...3....4.5.6.....6.3.....6...2...7.....8....4.2....9.....1.5.......7 #108 eleven`

I confirm that you found new unique vertical symmetric 18-clue pattern, having valid puzzle(s). Well done!

Serg

[Edited: I corrected name of symmetry considered in this thread - we consider vertical, not horizontal symmetry. Thanks to ronk for his correction.]
Last edited by Serg on Sun Jun 03, 2012 7:12 pm, edited 1 time in total.
Serg
2018 Supporter

Posts: 607
Joined: 01 June 2010
Location: Russia

### Re: Symmetric 18s

Great work eleven, on your discovery of #107 (earlier this year) and #108!

It's interesting both of these puzzles have exactly 2 clues per box.

They also both have a clue distribution by columns of 123 222 321 (or can be morphed that way).

#108 has a distribution of clues by rows of 123 222 222.

In other words, these last two sudokus have the clues spread out very uniformly. They aren't some strange freaky shapes - they are kind-of-boring (no 4 clues in a box, 6 clues in the center column, or anything like that).

Well, they aren't really boring! There's something mysterious that made them hide the best. Of all 108 these last two are my favorites

Good work!
olimpia

Posts: 35
Joined: 14 November 2008
Location: USA

### Re: Symmetric 18s

Hi olimpia,

nice to see, that you looked in again.

Now thats really bad luck. I got these mysterious magic numbers, i was an elected one, and then this:
Code: Select all
` +-------+-------+-------+ | . . . | . . . | . . . | | . . . | 1 2 3 | . . . | | . . 4 | . . . | 5 . . | +-------+-------+-------+ | . . . | 4 . 5 | . . . | | . 3 . | . . . | . 1 . | | 2 . . | . . . | . . 6 | +-------+-------+-------+ | . . 5 | . . . | 7 . . | | . . 7 | . 8 . | 9 . . | | . 1 . | . . . | . 2 . | +-------+-------+-------+............123.....4...5.....4.5....3.....1.2.......6..5...7....7.8.9...1.....2.  #109 eleven`
eleven

Posts: 1873
Joined: 10 February 2008

### Re: Symmetric 18s

Congratulations! on finding #109, especially so soon after #108.

eleven wrote:I dont understand, what you mean. Maybe you talk of 64 patterns with exactly 2 automorphisms, but i just meant, that all these puzzles have at least 2 automorphisms, simply because they are vertically symmetric. Obviously some patterns have an additional automorphism, because you can switch rows, but i did not count them.

Sorry, I misread "18=2*3*3 clue puzzles" as "18 puzzles" instead of "18 clues."

As to the number of automorphisms, "at least 2" is correct. However, strictly speaking it would be "at least 4." I raised this issue earlier and a consensus was not reached, and Pat agreed with you. To clarify, in vertically symmetric patterns there are two "validity preserving transformations", a) rotation about column 5, and b) swap of columns 4 and 6. Applying none, either one, or both yields 4 automorphisms.
ronk
2012 Supporter

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

### Re: Symmetric 18s

Yes, i also don't know where to look for a correct definition.
My personal wording is, that an automorphism is such a "validity preserving transformation". Depending on it's "cycle length", it can lead to (at least - again depending on a definition) 2 or 3 "equivalents" or (how i call them) "morphs".
So a puzzle with say 12 automorphisms in your sense may have 2 with cycle length 2 and 1 with cycle length 3 in my sense (all independant).
But maybe someone can point us to more official definitions.
eleven

Posts: 1873
Joined: 10 February 2008

### re:automorphisms of a vertically-symmetrical layout

ronk wrote:As to the number of automorphisms,
"at least 2" is correct.
However, strictly speaking it would be "at least 4."

I raised this issue earlier
and a consensus was not reached,
and Pat agreed with you.

To clarify, in vertically-symmetric patterns there are two "validity preserving transformations",
and b) swap of columns 4 and 6.
Applying none, either one, or both yields 4 automorphisms.

sorry about the phrasing of my Apr.17 post;
probably should've said --

each puzzle is part of a class of (at least) 4 isomorphs-with-the-same-layout
i.e. has (at least) 3 isomorphs-with-the-same-layout

. - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . -
automorphisms of a vertically-symmetrical layout --

• reflection around c5
does indeed bring us back to the same layout,
it (this reflection) is thus an automorphism of the layout.

• swapping c4<-->c6
likewise brings us back to the same layout,
and is thus a 2nd automorphism of the layout.

• applying both
may count as a 3rd automorphism of the layout
-- sorry i don't have the proper definitions, i'll let the real mathematicians clarify this.

• (but applying none -- the identity transformation -- is never counted as an automorphism.)

Pat

Posts: 3652
Joined: 18 July 2005

### Re: re:automorphisms of a vertically-symmetrical layout

Pat wrote:but applying none -- the identity transformation -- is never counted as an automorphism.)

I've never seen it not counted as an automorphism. Most puzzles are said to have 1 automorph, and the most canonical puzzle MC is said to have 648 automorphs. The only way those happen is to count the identity automorph. Perhaps its correct to not to count identity as an isomorph, but that would be inconsistent IMO.

You are a user of gsf's software, so just try "%#ac" in the format option and you'll see for yourself.
ronk
2012 Supporter

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

thanks, i stand corrected

Pat

Posts: 3652
Joined: 18 July 2005

### Re: Symmetric 18s

Hi, eleven!
eleven wrote:
Code: Select all
` +-------+-------+-------+ | . . . | . . . | . . . | | . . . | 1 2 3 | . . . | | . . 4 | . . . | 5 . . | +-------+-------+-------+ | . . . | 4 . 5 | . . . | | . 3 . | . . . | . 1 . | | 2 . . | . . . | . . 6 | +-------+-------+-------+ | . . 5 | . . . | 7 . . | | . . 7 | . 8 . | 9 . . | | . 1 . | . . . | . 2 . | +-------+-------+-------+............123.....4...5.....4.5....3.....1.2.......6..5...7....7.8.9...1.....2.  #109 eleven`

I checked your new pattern. It is really new vertical simmetric pattern. Congratulations!

I checked for minimality all 109 puzzles. They all are minimal.

How many vertical simmetric puzzles do you know? (Some time ago you published 385 vertical simmetric puzzles. What is their actual number now?)

Serg
Serg
2018 Supporter

Posts: 607
Joined: 01 June 2010
Location: Russia

### Re: automorphisms of a vertically-symmetrical layout

Hi, colleagues!
I think it's no sense to consider transformations "Swapping columns c4/c6" and "Reflection around column c5", because they act as "Do nothing" transformation for all considered patterns.

I propose to consider such patterns' isomorphic transformations:

1. Bands permutations (6 ways).
2. Rows permutations within the upper band (6 ways).
3. Rows permutations within the middle band (6 ways).
4. Rows permutations within the lower band (6 ways).
5. Correlated (mirrored) columns permutations within left and right stacks (6 ways).

So, alone pattern has at most 6^5 = 7776 automorphisms. And any vertically symmetric pattern should have at least one automorphism ("Do nothing" transformation).

Maybe it's worth to open separate thread for automorphisms discussion? I think this theme is rather far from discovering new 18-clue symmetric patterns.

Serg
Serg
2018 Supporter

Posts: 607
Joined: 01 June 2010
Location: Russia

PreviousNext